ECONOMICS 

OF 

ELECTRICAL  DISTRIBUTION 


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ECONOMICS 

OF  -:  ' 

ELECTRICAL  DISTRIBUTION 


BY 

P.  O.  REYNEAU,  M.E. 

MEMBER  A.I.E.E. 
ASSISTANT   ELECTRICAL    ENGINEER,    THE    DETROIT    EDISON    COMPANY 

AND 

H.  P.  SEELYE,  B.C.E. 

A8SO.    MEMBER  A.I.E.E. 
DISTRIBUTION    ENGINEERING    DEPARTMENT,    THE    DETROIT    EDISON    COMPANY 


FIRST  EDITION 


McGRAW-HILL  BOOK  COMPANY,  INC. 
NEW  YORK:  370  SEVENTH  AVENUE 

LONDON:  6  &  8  BOUVERIE  ST.,  E.  C.  4 
1922 


-  -  -  . 


COPYRIGHT,  1922,  BY  THE 
MCGRAW-HILL  BOOK  COMPANY,  INC. 


THE     MAPLE     PRESS     YORK    FA. 


PREFACE 

Electrical  Engineering  is  more  than  the  application  of  the 
principles  of  electricity  to  the  design,  construction  and  operation 
of  a  machine,  a  power  plant  or  a  distribution  system  to  the  end 
that  it  fulfills  satisfactorily  its  intended  purpose.  It  is  highly 
desirable  that  efficiency  be  also  accomplished.  Further  than 
this,  it  should  be  the  aim  of  all  engineers  to  develop  efficiency  in 
its  broader  sense,  that  is,  by  the  realization  of  maximum  economy 
in  both  design,  construction  and  operation. 

In  designing,  constructing,  or  operating  an  electrical  distribu- 
tion system,  the  object  to  be  striven  for  is  to  provide  all  customers 
with  a  good  quality  of  service  at  the  least  possible  cost  over  the 
system  as  a  whole.  This  result  can  be  attained  only  through 
a  careful  and  conscientious  application  of  the  principles  of 
economics  to  all  parts  of  the  system.  It  is  to  define  those 
principles  and  to  indicate  methods  for  their  application  that 
this  book  is  presented. 

This  work  has  been  prompted  by  the  realization  that  the  trans- 
mission and  distribution  system,  representing  a  large  part  of  the 
total  investment  in  any  central  station  and  offering  a  wide  field 
for  economic  study  has  not  often  been  given  sufficient  attention. 
Much  valuable  information  may  be  found  scattered  through  the 
engineering  literature,  but  it  is  thought  that  there  is  need  of 
bringing  together  the  factors  involved  in  such  a  study  and  of 
discussing  the  subject  as  a  whole. 

This  book  is  by  no  means  an  attempt  to  cover  the  whole  field 
of  economics  as  applied  to  the  central  station  system.  It  is 
well  recognized  that  the  range  of  problems  encountered  is  very 
broad  and  varied  and  that,  as  yet,  comparatively  little  progress 
has  been  made  in  such  work.  It  would  be  impossible  in  a  work  of 
this  kind  to  cover  in  detail  even  the  field  of  distribution  and  trans- 
mission economics.  The  purpose  of  the  book  is  to  present  the 
need  for  the  application  of  economic  principles  to  the  design  of 
distribution  systems,  to  explain  the  fundamental  principles  in- 
volved, to  indicate  the  types  of  problems  most  often  encountered 
and  to  offer  methods  of  studying  such  problems  and  reaching 
their  solution. 


<: 


vi  PREFACE 

A  considerable  part  of  the  material  presented  in  this  book 
has  appeared  previously  in  the  Electrical  World.  The  chapter 
on  single-phase  secondaries  is  a  rearrangement  of  material 
published  in  the  Proceedings  of  the  American  Institute  of  Elec- 
trical Engineers. 

Merrill  W.  De  Merit  has  furnished  valuable  assistance  par- 
ticularly in  the  chapters  on  Energy  Cost  and  Underground 
Lines. 

DETROIT,  MICH.,  THE  AUTHORS. 

Dec.  1921. 


CONTENTS 

PAGE 

PREFACE v 

CHAPTER 

I.  INTRODUCTORY 1 

II.  APPLICATION  OF  ENGINEERING  ECONOMICS  TO  TRANSMISSION  AND 

DISTRIBUTION  PROBLEMS 5 

III.  COSTS 11 

Principles  underlying  the  determination  of  true  costs — 
Formulas  for  unit  labor  costs — Cost  records — Annual  costs. 

IV.  ENERGY  COST 22 

Principles  and  methods  involved  in  the  determination  of 
the  cost  of  energy  and  of  energy  losses. 

V.  LOAD  CHARACTERISTICS 37 

Power  factor — Balance  factor — Demand  factor — Diversity 
factor — Load  factor — Equivalent  hours. 

VI.  GENERAL  EQUATIONS 48 

Kelvin's  law — General  method  of  solving  problems — Presen- 
tation of  results. 

VII.  POWER  Loss  AND  VOLTAGE  DROP 55 

Charts   for    simplified    solution  for  power  loss  and  voltage 
drop. 
VIII.  TRANSMISSION   LINE   PROBLEMS — "BACKBONE"    TRANSMISSION 

LINES 71 

Method  of  determining  most  economical  design  for  main  or 
"backbone"  transmission  lines. 

IX.  TRANSMISSION    LINE     PROBLEMS — SECONDARY     TRANSMISSION 

LINES  . .    .    .  • 81 

Determination  of  most  economical  standards  of  construction, 
conductor  size,  loading,  route,  etc.  on  lesser  or  secondary 
transmission  lines. 

X.  RECONSTRUCTION  PROBLEMS 96 

Principles  involved  in  the  solution  of  problems  dealing  with 
the  alteration  or  reconstruction  of  lines  already  built — 
Method  of  including  value  of  salvaged  material  in  cost  study. 

XI.  POWER  CIRCUITS 109 

Problems  relating  to  lines  carrying  power  load  chiefly — 
Voltage — Economical  conductor  size — Use  of  two  lines  in 
place  of  one — Distribution  of  load  over  several  lines. 

XII.  LIGHTING  CIRCUITS :    .    .    128 

Economical  studies  on  circuits  carrying  lighting  only — Pre- 
diction of  load — Conductor  size — Increasing  capacity  of 
overloaded  systems. 

vii 


viii  CONTENTS 

XIII.  SECONDARY  DISTRIBUTION — SINGLE  PHASE 139 

Study  of  most  economical  design  for  secondaries — Voltage 
drop  —  Conductor  size  —  Transformer  size — Length  of 
secondary. 

XIV.  POWER  SECONDARIES 174 

Power  secondary  vs.  separate  transformers — Economical  size 
of  3</>  secondaries. 

XV.  UNDERGROUND  LINES 183 

Voltage — Cable  size — Route — Number  of  ducts  in  a  duct 
line — Arrangement  of  ducts  and  cables. 

XVI.  THE  SYSTEM  AS  A  WHOLE 194 

XVII.  INDUSTRIAL   PLANT  PROBLEMS 197 

Application  of  industrial  plant  problems  of  the  principles 
of  economics  outlined  for  electrical  distribution. 

APPENDIX 201 

INDEX.  .   207 


ECONOMICS  OF 
ELECTRICAL  DISTRIBUTION 

PART  I 

CHAPTER  I 
INTRODUCTORY 

Electrical  engineering  design  has  three  necessary  components, 
i.e.  electrical  design,  structural  design  and  economical  design. 
They  are  equally  important  and  no  electrical  engineering  problem 
can  be  solved  properly  without  consideration  of  all  three  compo- 
nents in  a  thorough  manner.  The  solution  of  any  problem  will 
come  nearest  to  perfection  as  each  of  these  forms  of  design  is 
more  soundly  analysed  and  the  combination  of  the  three  more 
intelligently  applied. 

Structural  design  calls  for  the  study  of  materials  used  and  their 
combination  into  the  desired  structures  in  such  a  way  that  proper 
factors  of  safety  may  be  obtained.  Electrical  design  calls  for  a 
knowledge  of  electrical  phenomena  and  the  application  of  this 
knowledge  to  achieving  results  satisfactory  from  an  operating 
standpoint.  Economical  design  calls  for  the  knowledge  of  costs 
and  their  application  to  the  determination  of  the  most  economical 
design  possible.  It  is  evident  that  there  must  be  overlapping 
between  these  three  fields.  Structural  design  depends  in  most 
cases  on  the  size  of  electrical  apparatus,  conductors,  etc.,  which 
in  turn  is  governed  by  the  electrical  design.  Both  structural 
and  electrical  features,  on  the  other  hand,  should  be  planned  with 
a  view  toward  economy.  Where  there  is  a  choice  of  more  than 
one  possible  design  which  is  satisfactory  from  a  structural  and 
an  electrical  standpoint,  the  decision  should  be  based  on  study  of 
the  relative  economy  of  all  the  alternatives  considered.  It  is  pro- 
posed in  this  book  to  consider  electrical  distribution  from  the 
standpoint  of  economy  principally.  There  is  no  intention 

1 


2  ECONOMICS  OF  ELECTRICAL  DISTRIBUTION 

however  of  minimizing  the  importance  of  structural  and  elec- 
trical design.  Economical  design  has  probably  received  less 
attention  than  the  two  other  forms  and  the  emphasis  brought  on 
it  here  may  help  to  give  it  its  proper  place  in  the  work  of  obtaining 
correct  solutions  to  electrical  distribution  problems. 

Scientific  books  in  general  may  be  divided  into  two  classes. 
The  first  class  gives  facts  and  data  obtained  from  practical  experi- 
ence. The  second  class  gives  methods  and  suggestions  of  means 
of  attacking  the  study  of  certain  problems.  This  book  belongs 
to  the  second  class.  In  work  with  distribution  systems  the 
conditions  met  with  in  different  localities  are  diverse.  Costs  of 
material  and  labor  and  methods  of  construction  and  operation 
are  varied  and  changeable.  It  follows  that  if  definite  figures 
and  results  are  given  which  might  apply  to  one  system  at  one 
particular  time,  these  same  figures  might  be  very  misleading  if 
applied  indiscriminately  to  any  other  locality,  system  or  period  of 
time.  It  is  necessary,  therefore,  that  any  figures  given  here, 
particularly  those  referring  to  costs,  should  be  considered  only  as 
examples  of  the  methods  presented  and  not  as  material  on  which 
to  base  any  calculations. 

Furthermore,  there  is  no  intention  to  cover  fully  either  in 
detail,  or  in  a  general  descriptive  manner  all  the  problems,  which 
might  be  encountered  in  electrical  distribution,  viewed  from  the 
economic  standpoint.  It  is  desired  to  present  some  general 
methods  that  can  be  applied  to  the  solution  of  most  of  these 
problems,  to  give  examples  of  the  application  of  these  principles 
to  some  ordinary  cases,  and  to  indicate  the  types  of  questions 
most  often  arising  in  this  work. 

The  book  is  made  up  of  two  parts.  The  first  part  including 
Chapters  II  to  VII  is  intended  to  give  working  methods,  or  so  to 
speak,  to  furnish  the  tools  to  be  used  in  solving  electrical  dis- 
tribution problems  economically.  The  authors  have  tried  to 
present  the  underlying  principles  which  are  the  basis  of  economic 
study  of  this  kind.  The  fundamentals  are  not  new  but  their 
application  to  the  design  of  a  distribution  system  is  not  so 
generally  understood.  The  use  of  accurate  annual  costs  instead 
of  first  costs  (or  loose  estimates)  as  a  basis  for  determining  the 
most  economical  installation  goes  back  to  Lord  Kelvin  and 
further.  But,  we  see  many  engineers  today  ignoring  the 
economical  factors  of  the  problem  entirely  and  others  basing  all 
comparisons  on  first  cost  only,  or  attempting  to  apply  the 


INTRODUCTORY  3 

so-called  Kelvin's  Law  indiscriminately.  A  knowledge  of  the 
fundamentals  is  absolutely  essential  if  any  such  study  is  to  be 
worth  the  time  spent  upon  it.  Otherwise  the  engineer  may  be 
guilty  of  deluding  himself  and  others  with  figures  which  are 
entirely  inapplicable  to  the  case  in  hand. 

Chapter  II  attempts  to  create  a  point  of  view  regarding  the 
subject  at  hand.  Then  in  succession  are  taken  up  means  of 
analyzing  and  tabulating  costs  of  material  and  labor,  of  deter- 
mining the  cost  of  energy  to  be  used  under  various  conditions, 
of  studying  the  characteristics  of  loads  to  be  handled,  and  of 
forming  general  equations  for  solving  problems  in  economical 
design.  Finally  a  chapter  on  power  loss  and  voltage  drop  is 
included  to  provide  convenient  means  of  handling  these  most 
important  electrical  phenomena  which  must  always  be  considered 
in  connection  with  economical  design. 

The  second  part  of  the  book  consists  in  presenting  the  applica- 
tion of  the  methods,  or  the  use  of  the  tools  described  in  the  first 
part.  For  convenience  a  division  of  the  subject  has  been  made 
into  transmission  lines,  power  circuits,  lighting  circuits,  second- 
aries and  underground  lines.  Under  each  heading  some  of  the 
general  problems  encountered  are  indicated  and  some  particular 
problems  are  solved  in  detail  as  definite  examples.  Special  or 
unusual  problems  have  not  been  considered,  the  work  for  the 
most  part  holding  to  the  everyday  questions  met  with  in  practice. 
Chapter  XVI  in  conclusion,  touches  on  a  few  of  the  general  prob- 
lems which  apply  to  the  system  as  a  whole,  such  as  the  location 
of  generating  station  and  substations,  the  relation  of  one  part  of 
the  system  to  the  remainder,  etc.  Chapter  XVII  takes  up 
briefly  a  few  of  the  problems  in  distribution  pertaining  to 
industrial  plants. 

It  will  be  seen  that  the  possibilities  for  study  in  the  field  of 
distribution  economics  are  boundless.  The  deeper  one  goes  into 
the  subject  the  more  problems  present  themselves  and  the  more 
evident  becomes  the  need  for  careful  and  accurate  investigation. 
The  objection  may  be  raised  that  the  changeable  character  of 
the  loads  carried,  and  the  fluctuations  of  prices  make  a  too 
detailed  study  of  economies  impracticable.  That  is  no  doubt 
true  sometimes,  as  regards  individual  problems.  In  the  long 
run,  however,  the  knowledge  gained  is  always  valuable  and  of 
great  assistance  in  deciding  general  policies.  As  our  information 
on  the  behavior  of  materials  and  on  the  characteristics  of  the 
various  types  of  loads  becomes  more  accurate,  the  practicability 


4  ECONOMICS  OF  ELECTRICAL  DISTRIBUTION 

of  the  application  of  detailed  economies  will  increase.  With  the 
increasing  demand  for  efficiency  and  economy  in  all  lines  of 
endeavor,  the  necessity  for  the  economical  operation  of  distri- 
bution systems  becomes  more  imperative.  In  any  case,  an  exact 
knowledge  of  the  true  cost  will  always  lead  to  efforts  to  reduce 
that  cost  if  possible. 


CHAPTER  II 

APPLICATION     OF     ENGINEERING     ECONOMICS     TO 
TRANSMISSION  AND  DISTRIBUTION  PROBLEMS 

As  stated  in  the  first  chapter,  the  purpose  of  this  book  is  to 
present  some  of  the  problems  encountered  in  the  design  of  lines 
for  transmitting  electrical  energy  and  to  approach  their  solution 
from  an  economic  point  of  view.  More  specifically  it  is  purposed 
to  apply  engineering  economics  to  distribution-  and  transmission- 
line  layouts  indicating  how  the  most  economical  installation 
consistent  with  good  service  can  be  determined. 

No  argument  should  be  required  as  to  the  advisability,  or 
rather  necessity,  for  the  application  of  economic  principles  to 
engineering  design.  Engineering  should  make  for  efficiency  but 
there  can  be  no  real  efficiency  unless  there  also  is  accomplished 
economy.  Economy  cannot  often  be  recognized  at  first  glance. 
A  working  knowledge  of  its  fundamentals  at  least  is  required. 
Engineering  and  engineering  economics  are  therefore  synonymous 
in  all  problems  directly  involved  in  the  production  or  distribution 
of  a  commercial  commodity.  This  includes  the  great  majority 
of  our  present  everyday  engineering  problems  and  naturally 
those  of  transmission  and  distribution. 

It  is  not  within  the  scope  of  this  book  to  consider  unusual 
problems  such  as  very  high-voltage  transmissions  for  example,  or 
installations  of  special  apparatus  whose  use  has  not  become  a 
matter  of  accepted  practice.  Rather,  the  general  purpose  will 
be  to  make  a  few  analyses  that  will  indicate  methods  of  finding 
the  most  economical  design  under  usual  conditions  with  material 
of  known  characteristics  and  at  known  prices.  It  is  especially 
for  the  improvement  of  the  design  of  the  large  class  of  everyday 
jobs  that  this  work  is  written.  Such  analyses  will,  naturally, 
not  only  indicate  the  most  economical  installation  with  materials 
and  construction  methods  already  in  use  but  will  make  it  possible 
to  determine  where  any  economy  can  be  effected  by  changes  in 
these  methods  or  materials.  It  must  be  recognized  that  the 

5 


6  ECONOMICS  OF  ELECTRICAL  DISTRIBUTION 

majority  of  the  transmission  and  distribution  lines  have  been 
laid  out  by  rule-of -thumb  method,  with  no  definite  conception 
of  actual  costs  or  economy.  The  methods  here  presented  have 
been  developed  by  the  study  of  actual  problems  encountered  in 
practice.  It  is  hoped  that  these  methods  will  on  the  one  hand 
show  the  magnitude  of  the  savings  possible  by  the  use  of  engineer- 
ing economics  and  on  the  other  hand  be  simple  enough  in  their 
application  to  be  considered  worth  using  in  many  places  where 
more  empirical  methods  are  used  today. 

METHOD  OF  TREATMENT 

Practically  all  such  problems  of  engineering  design  require 
treatment  in  three  distinct  stages  for  a  complete  solution.  At 
first  all  the  data  must  be  obtained  that  can  be  determined  with 
exactness.  Such  elements  as  material  costs,  strength  of  parts  of 
construction,  load  tests,  etc.,  are  here  included.  Second,  those 
elements  which  are  not  subject  to  exact  measurement  or  compu- 
tation must  be  decided  upon.  These  are  usually  a  development 
from  practical  experience  over  a  number  of  years  and  are  often 
matters  of  accepted  standard  practice.  They  are  such  items  as 
probable  increase  in  load,  allowable  voltage  regulation,  safety 
factors,  standards  of  construction,  etc.  Empirical  methods  must 
here  be  employed.  The  third  element  is  the  most  intangible — 
the  good  judgment  of  the  engineer  based  on  his  knowledge  and 
experience  and  applied  to  the  particular  problem  in  hand  to  so 
utilize  the  first  two  elements  as  to  create  an  efficient  and  econom- 
ical design  best  adapted  to  the  situation.  All  these  elements  in 
their  proper  proportions  are  equally  important  in  attaining  the 
most  satisfactory  solution  for  any  p'roblem..  Empirical  methods 
if  applied  blindly  may  lead  far  away  from  the  purpose  for  which 
they  were  originally  intended.  Judgment  based  on  experience 
alone  is  liable  to  become  a  mere  guess  unless  supported  by  exact 
knowledge  and  good  practice.  It  would  appear  therefore,  that 
the  further  the  exact  data  can  be  carried  in  a  problem  the  less 
dependence  need  be  placed  on  the  more  intangible  elements  and 
hence  the  greater  certainty  of  the  best  solution.  The  extension 
of  this  exact  knowledge  is  the  chief  purpose  of  the  study  of  the 
economic  features  of  a  problem.  In  many  cases  economy  may 
be  made  the  deciding  factor  between  two  or  more  designs  appar- 
ently equally  good  from  other  points  of  view. 


APPLICATION  OF  ENGINEERING  ECONOMICS  7 

Exact  Data. — For  any  given  system  there  are  a  number  of 
limiting  conditions  which  reduce  the  unknown  factors  in  the 
design.  There  are  certain  standards  such  as  transformer  sizes, 
wire  and  cable  sizes,  etc.,  which  are  established  by  the  manufac- 
turer. Other  limitations  are  established  by  the  accepted  good 
practice  of  the  profession  in  general.  Also  each  company  has 
certain  standards  of  materials  and  construction  to  which  it  is 
wise  to  adhere  unless  there  is  a  clear  advantage  in  making  a 
change.  The  most  complete  information  obtainable  on  all  such 
data  relating  to  the  particular  system  in  hand  should  be  kept 
available  for  ready  reference.  Also  other  items  which  can  be 
obtained  exactly  for  only  the  one  locality  or  company  such  as 
costs  of  material,  labor  and  energy  should  be  determined  as 
accurately  as  possible  and  be  revised  from  time  to  time  to  con- 
form to  changing  prices,  wage  scales  and  costs  of  production. 
This  information  can  be  readily  kept  up  to  date  and  tabulated  or 
drawn  up  in  curves.  Then  only  such  factors  as  are  inherent  to 
the  particular  line  itself  need  be  determined  in  making  the 
design. 

Empirical  Data. — Some  of  the  empirical  elements  of  the 
problem  of  transmission  and  distribution  of  energy  are  worthy 
of  considerable  attention.  One  of  these  is  the  element  of  "good 
service."  It  is  understood  that  in  any  design  or  layout  the 
greatest  economy  " consistent  with  good  service"  is  the  ob- 
ject. Just  what  is  good  service  may  be  a  matter  of  considerable 
question. 

Good  Service. — Good  service  is  not  dependent  on  the  ideas  of 
the  engineer  or  the  "server"  as  to  what  it  should  be.  Service  is 
good  when  it  satisfies  the  one  that  is  served.  That  man  who  at 
the  receipt  of  his  bill  feels  that  he  has  gotten  all  that  was  due 
him  is  receiving  good  service.  He  is  willing  to  exchange  his 
money  for  what  he  has  received  from  the  server. 

The  human  element  is  therefore  the  most  important  one  in 
determining  good  service.  Customers  will  expect  as  good  or 
better  operating  conditions  than  they  have  been  in  the  habit  of 
getting.  Companies  have,  so  to  speak,  educated  their  public 
to  certain  expectations  and  they  must  live  up  to  such  expecta- 
tions. It  is  evident  however  that  it  is  impossible  to  eliminate  all 
interruptions  or  conditions  of  poor  regulation.  In  thickly 
populated  districts  with  a  high-load  density,  practically  continu- 
ous service  at  good  voltage  is  easily  given.  In  outlying,  scattered 


8  ECONOMICS  OF  ELECTRICAL  DISTRIBUTION 

districts  the  cost  of  the  same  quality  of  service  would  be  pro- 
hibitive. From  the  company's  point  of  view  the  service  must  be 
of  good  enough  quality  to  insure  earnings  and  yet  not  so  good  as 
to  necessitate  too  large  expenditures.  It  is  therefore  necessary, 
before  attempting  to  solve  a  transmission  or  distribution  problem 
economically,  to  ascertain  what  quality  of  service  is  required 
both  from  the  customer's  and  the  company's  point  of  view. 

Good  service  can  therefore  only  be  determined  in  a  general 
way  and  will  vary  with  locality,  company,  kind  of  load,  etc. 
It  might  be  said,  however,  that  with  the  modern  refinement  of 
methods  and  equipment  and  the  nearly  universal  high  require- 
ment of  the  customer  we  are  approaching  more  and  more  a 
condition  when  wide  variation  of  voltage  and  long  or  frequent 
interruptions  will  hardly  be  permissible. 

Increase  in  Load. — A  second  quantity  which  must  be  empiri- 
cally determined  is  that  of  load  increase.  The  variation  of  the 
loads  to  be  carried  and  the  growth  of  energy  demand  that  is 
prevalent  in  nearly  all  localities  are  always  matters  for  consider- 
able study  in  connection  with  a  layout.  There  are  some  cases 
where  it  is  possible  to  design  for  a  certain  demand  that  can  be 
assumed  to  remain  constant  for  a  period  of  years,  possibly  for  the  life 
of  some  part  of  the  equipment.  Such  would  be  the  case  in  planning 
lines  to  a  large  power  installation  where  growth  could  be  taken 
care  of  by  new  circuits.  On  the  other  hand  such  load  as  house 
lighting  increases  continuously  and  its  increase  can  only  be  esti- 
mated from  the  figures  for  the  past  few  years  and  general  business 
conditions.  In  any  case  we  must  design  for  a  system  that  will 
be  most  economical  over  its  useful  life,  in  other  words  when  the 
sum  of  the  annual  costs  for  all  years  under  consideration  will  be  a 
minimum.  It  is  further  necessary  to  study  as  exactly  as  possible 
the  number  of  years  to  be  cared  for  by  the  present  design  and 
minor  changes  that  can  be  made  at  various  times  during  that 
period  for  taking  care  of  the  changes  in  load.  Generally  the 
design  should  cover  the  expected  life  of  that  part  of  the  equipment 
whose  life  is  shortest  and  which  would  require  a  large  expenditure 
for  its  replacement.  Any  considerable  change  necessary  to 
care  for  larger  load  could  then  be  made  at  the  same  time  with 
relatively  smaller  cost.  In  some  cases,  however,  where  the 
probable  increase  in  load  is  more  definitely  known,  the  economical 
life  of  the  present  design  can  be  accurately  determined  even 
well  within  the  expected  life  of  all  the  important  parts. 


APPLICATION  OF  ENGINEERING  ECONOMICS  9 

Financial  Conditions. — The  condition  of  the  money  market 
may  be  another  determining  factor  in  an  economical  design. 
Often  the  difficulty  in  securing  necessary  funds  for  a  job  may 
require  the  engineer  to  spend  less  at  the  time  of  installation  than 
is  consistent  with  economy  over  the  period  the  design  covers. 
However  the  condition  should  be  carefully  considered  in  order 
to  still  obtain  the  best  design  as  limited  by  the  money  available. 

Relation  of  Parts  to  Whole  System. — The  fact  that  the  trans- 
mission or  distribution  line  is  only  one  part  of  the  system  trans- 
mitting energy  from  the  turbine  to  the  customer  must  be  kept  in 
mind.  Its  design  will  affect  and  be  affected  by  existing  or 
planned  conditions  in  the  other  parts.  It  is  therefore  always 
necessary  to  treat  a  line  not  only  as  an  independent  unit  for 
some  specific  purpose  but  also  as  a  working  part  of  the  whole 
system. 

Bearing  in  mind  the  general  considerations  brought  out  above, 
the  table  given  below  will  show  the  most  important  points  to 
consider  in  studying  the  layout  of  an  economical  line.  The 
following  chapters  will  discuss  in  more  detail  the  methods  of 
obtaining  this  data  and  its  application  to  particular  classes  of 
problems  such  as  transmission  lines,  power  lines,  secondaries, 
etc. 


1.  Load 


2.  Route  of  line 


Location. 

Present  size. 

Probable  increase  and  rate  of  increase. 

Characteristics:  Phase  requirements. 

Power  factor. 

Variation,  daily,  weekly,  seasonal,  etc., 
and  relation  to  station  variations  and 
peak. 

Unbalance  factor. 

Maximum. 

Load  factor. 

Investigation  of  advisability  of  possible  routes,  affected 
by :  Available  pole  or  duct  space. 
Purchase  of  right-of-way. 
Difficulties  in  construction. 
Interference  with  or  from  other  lines,  physically 

or  electrically. 
Convenience  of  operation. 
Possible  future  extensions. 
Effect  on  system  as  a  whole. 
Cost  as  compared  with  other  possible  routes. 


10 


ECONOMICS  OF  ELECTRICAL  DISTRIBUTION 


3.  Physical 
constants 


4.  Costs 


5.  Relation  to 
System 


Length  of  line 
Spacing 
Volfcage:  desired, 

available. 
Resistance  of  line 
Reactance 
Voltage  drop 
Regulation:  desired, 
available 


Under  various  line  conditions  as 
limited 

1.  By  good  operation. 

2.  By  standard  practice. 

3.  By  routing. 

4.  By  economical  considerations. 

5.  By  good  service. 


1.  Determination  of 
labor  costs, 

material  costs,  and  |  For    all    conditions    considered 
energy  loss  costs      ( 

2.  Application  of  those  costs  to  determine  the  best  line  for 
the  purpose. 

General  considerations  that  may  affect  most  economical 
line  found  above  under  (4)  due  to  its  being  a  part  of  a 
large  system. 
Financial  considerations. 


From  inspection  of  this  table  it  is  seen  that  the  five  divisions 
in  it  are  in  one  way  or  another  interdependent  so  that  each  one 
must  be  considered  in  relation  to  all  the  others.  It  is  only  by  a 
careful  study  of  the  problem  from  all  angles  and  a  coordination 
of  results  that  a  satisfactory  solution  can  be  obtained.  In  this 
book,  however,  the  economic  features  of  the  design  have  been 
emphasized  rather  than  the  mechanical  or  electrical.  These 
latter  have  been  covered  quite  thoroughly  in  other  works. 


CHAPTER  III 
COSTS 

PRINCIPLES  UNDERLYING  THE  DETERMINATION  OF  TRUE  COSTS — 
FORMULAS  FOR  UNIT  LABOR  COSTS — COST  RECORDS — 

ANNUAL  COSTS 

The  fundamental  basis  underlying  the  whole  question  of  econo- 
mic design  is  the  accurate  determination  of  costs,  i.e.,  costs  of 
construction  and  cost  of  energy.  If  these  are  not  correctly 
determined  any  conclusions  drawn  from  their  use  will  have  little 
value.  Further  it  is  generally  inadvisable  to  accept  for  this 
purpose  any  cost  data  which  has  not  been  locally  derived,  as 
every  power  company  has  its  own  methods  and  standards  of 
construction,  its  own  labor  costs,  its  own  efficiencies  of  operation. 
These  may  differ  widely.  Hence,  it  is  absolutely  necessary 
to  make  as  complete  a  determination  of  local  costs  as  possible  as 
a  basis  for  any  economic  study. 

Evidently  when  it  is  decided  to  establish  a  cost  record  on  con- 
struction, material  and  labor,  difficulties  will  be  encountered  in 
obtaining  correct  information  regarding  all  details.  Much 
valuable  information  will  be  found  in  appraisals  and  much  can 
be  gotten  from  the  accounting  records  by  a  skillful  economist. 
However,  in  most  cases  it  will  take  several  years  to  establish  a 
complete  record  and  special  studies  will  be  necessary.  This 
record,  when  obtained,  is  an  easy  thing  to  keep  up  to  date  and 
in  convenient  form  for  use.  In  the  meantime  certain  make- 
shifts will  be  necessary.  If  a  complete  record  of  unit  construc- 
tion and  operating  costs  is  not  obtainable  or  when  lack  of  time 
or  of  available  records  requires  any  item  of  cost  to  be  estimated, 
the  estimate  should  be  based  on  known  conditions  as  far  as 
possible.  Further  than  this  there  should  be  determined  by  what 
percentage  the  final  result  will  be  affected  by  any  reasonable 
variation  in  the  assumed  cost.  For  example,  the  cost  of  energy 
at  any  point  may  be  assumed  to  be  1  ct.  per  kilowatt-hour.  If 
this  is  not  an  accurately  determined  figure  however,  it  would  be 
well  to  determine  how  the  result  would  be  affected  if  the  cost  of 
energy  were  say  %  ct.  or  1J4  cts.  per  kilowatt-hour.  Such  a 
comparison  will  at  least  place  the  solution  of  the  problem  within 
definite  limits. 

11 


12  ECONOMICS  OF  ELECTRICAL  DISTRIBUTION 

It  is  a  generally  recognized  fact  that  the  computation  of  first 
cost  is  only  a  step  in  the  investigation  of  the  real  cost  of  a  proposi- 
tion. The  economy  in  comparison  with  any  other  alternative 
proposition  can  only  be  determined  when  the  annual  cost  is 
investigated.  The  annual  cost  must  include  all  annual  charges 
against  the  investment  itself  and  all  operating  costs,  maintenance, 
repair  and  energy  losses.  Of  course  annual  costs  cannot  always 
be  considered  the  determining  factor,  since  on  items  involving 
large  expenditure  the  possible  difficulty  of  obtaining  capital 
may  have  considerable  weight.  Usually,  however,  annual  cost 
may  be  accepted  as  the  criterion. 

Before  annual  costs  on  a  piece  of  property  such  as  a  trans- 
mission line  can  be  obtained  it  is  necessary  to  determine  its  total 
value  or  first  cost. 

First  Cost. — The  determination  of  cost  figures  for  general  use 
is  greatly  facilitated  if  line  construction,  materials  and  methods 
are  standardized.  This  allows  a  fairly  accurate  figure  to  be 
obtained  for  standard  units  as  per  pole,  or  per  1,000  ft.  or  per 
mile  for  any  type  of  construction.  Otherwise  smaller  units  must 
be  depended  upon  such  as  per  crossarm,  per  insulator,  per  100 
ft.  of  wire,  etc.  In  any  case  it  will  be  found  exceedingly  valuable 
to  have  as  complete  a  record  as  possible  of  itemized  costs  from 
the  smallest  part,  such  as  a  bolt,  up  to  an  average  cost  for  a  large 
assembled  unit  such  as  per  mile  of  line.  This  should  include 
both  material  and  labor  costs  with  overhead  expense  all  shown 
separately.  Provision  should  also  be  made  for  easy  revision  of 
costs  as  prices  change.  This  revision  should  be  frequently  made 
when  prices  of  labor  and  material  are  fluctuating  to  any  extent 
since  a  large  part  of  economic  study  deals  with  possible  new 
construction  which  will  be  at  present  or  future  prices  or  with  old 
construction  which  represents  a  value  equal  to  new  construction 
less  a  certain  percentage  for  its  age.  Certain  types  of  problems 
require  the  consideration  of  the  actual  cost  of  the  old  construction 
at  the  time  it  was  installed  but  these  are  not  so  general. 

There  are  a  number  of  items  which  must  be  included  in  addi- 
tion to  current  quotations  on  material  or  actual  labor  time  in 
erecting.  To  material  price  may  be  added  such  items  as  freight, 
treating  material  for  poles,  tie  wires  on  line,  etc.  To  actual 
unit  labor  costs  should  be  added  a  proportional  amount  for 
unoccupied  time,  rainy  days,  vacations,  transportation 
inspection,  etc.  For  example: 


COSTS 


13 


Weight 

Cost  at 

Freight 

Injury 

Ties 

Total 

Per  cent 

per 

21  cts. 

on  re- 

to reels 

per  mile 

per  mile 

of  wire 

mile, 

per 

turned 

cost 

No.  2  Solid  wire 

one 

pound 

reels 

wire, 

pounds 

1,066  Ib. 

$223.86 

1.86 

2.32 

3.63 

$231.67 

103.4 

Naturally  these  additional  items  vary  for  each  unit  and  each 
locality.  It  is  usually  possible  to  obtain  a  percentage  which 
may  be  added  to  actual  price  to  include  such  incidental  material. 
Loading. — The  item  of  overhead  expense  or  loading  is  one 
which  must  be  included  in  nearly  all  problems  involving  costs. 
There  are  various  methods  of  applying  this  loading  but  for  the 
purpose  under  consideration  the  method  of  unit  loading  is 
probably  the  most  satisfactory,  i.e.,  apportioning  to  each  indi- 
vidual item  its  pro  rata  cost  for  different  items  of  overhead 
expense,  both  on  material  and  labor.  It  is  evident,  for  example, 
that  the  item  of  breakage  will  be  greater  for  insulators  than  for 
wire.  The  items  which  may  legitimately  be  included  in  loading 
are  as  follows: 

Waste — end  trimmings  of  wire,  cable  cut  back  for  splicing,  incompetent 
labor,  etc. 

Loss  and  breakage — theft,  broken  insulators,  etc. 

Tool  expense — tools  used  up,  broken,  stolen  and  repairs  and  depreciation 
on  them. 

Direct  supervision — engineering,  heads  of  departments,  general  foremen, 
office  expense,  clerks,  stenographers. 

Injuries  and  damages — doctor  and  hospital  expense,  liability,  insurance, 
tc. 

Purchasing  expense — all  expenses  of  purchasing  department. 

Stores  and  supplies  expense — all  expenses  of  stores  department  and 
handling. 

The  percentage  to  be  applied  to  any  unit  to  cover  any  one  of 
the  above  items  will  depend  entirely  on  local  conditions  and  no 
figures  could  be  given  here  which  would  be  of  any  value.  The 
total  loading  percentage  is  usually  between  10  and  25  per  cent 
depending  upon  the  class  of  property  and  upon  local  conditions. 

As  will  be  seen  later  all  problems  do  not  require  a  detailed 
application  of  loading.  Many  problems  involving  the  compari- 
son of  relatively  small  amounts  of  construction  may  be  safely 
considered  from  the  point  of  actual  material  and  labor  costs 


14  ECONOMICS  OF  ELECTRICAL  DISTRIBUTION 

alone,  considering  overhead  to  be  equal  in  both  cases.  The 
question  of  whether  or  not  to  apply  loading  must  be  determined 
by  the  conditions  involved  in  the  particular  problem  under 
consideration.  In  general  where  good  cost  records  are  kept 
it  will  be  just  as  easy  and  more  accurate  to  include  it  in  all 
problems. 

Labor  costs  on  any  unit  may  usually  be  reduced  to  an  equiva- 
lent formula  of  man  hours  for  various  classes  of  labor  as  foreman, 
lineman,  groundman,  etc.,  for  each  unit  plus  a  percentage  for 
incidental  expenses  as  above.  Such  formulas  facilitate  revision 
of  prices  when  necessary  as  the  current  wages  may  be  substituted 
and  the  labor  cost  on  the  unit  easily  obtained. 

For  example,  if  a  gang  consisting  of  1  foreman,  1  truck,  1 
chauffeur,  4  linemen  and  3  groundmen  can  string  1.6  miles  of 
No.  0  bare  stranded  wire  per  day,  on  an  average,  the  cost  per 
mile  of  wire  will  be  .728  times  the  cost  of  the  gang  per  day 

+  IT  +  ICh  +  4L  +  3Gr) 


where  the  symbols  represent  the  daily  wage  of  the  various 
classes  of  labor  included,  foreman,  truck,  chauffeur,  linemen  and 
groundmen. 

Some  of  the  labor  formulas  are  more  complicated  but  all  are 
computed  on  the  same  basis.  For  poles,  for  example,  several 
different  gangs  are  included  in  the  labor  charge,  i.e.,  unloading, 
distribution,  framing  and  roofing,  digging  hole,  and  setting  and 
the  gangs  for  each  operation  may  be  different  for  different  sizes 
of  poles.  A  formula  similar  to  the  above  may  be  determined  for 
each  operation  and  the  total  labor  formula  for  the  pole  is  a 
composite  of  these.  For  example: 

if  Gi  represents  the  daily  cost  of  the  unloading  gang  =(17V  -f  ICh  +  3GV), 

Ga  represents  the  daily  cost  of  the  distributing  gang  =(17V  +  ICh  +  3(?r), 

Gi  repres  nts  the  daily  cost  of  framing  and  roofing  =(HL  +  2Gr), 

Gi  represents  the  daily  cost  of  the  digging  gang  =  (H  F  +  MTr  +  MCh  +  IGr), 

G&  represents  the  daily  cost  of  the  setting  gang  =(6<,F-\-%Tr+tfCh+5Gr) 

(The  H  f,  %  Tr,  etc.,  are  occasioned  by  the  fact  that  one  foreman  and 
one  truck  serve  several  gangs  at  one  time.) 

Then  the  total  labor  cost  on  one  35-ft.  pole  equals 

.OlGOGj  +  .1403G2  +  .063G3  +  .2243£4  +  .0468G5 

Cost  Records.  —  For  a  complete  record  of  line  costs  there  are 
necessary  the  following  items: 


COSTS 


15 


(a)  Current  prices  of  material  and  labor  of  all  kinds. 
(6)   Labor   formulas   and  constant  multipliers  for  various  classes    of 
materials  with  loading  percentages  for  both. 

(c)  Current  material  and  labor  costs  on  units  of  construction,  as  per  cross- 
arm,  per  pole,  per  insulator,  etc. 

(d)  Current  costs  on  assemblies.     Assemblies  may  range  from  small  items 
such  as  a  crossarm  erected  with  braces,  bolts,  etc.,  or  a  ground  connection 
with  wire,  ground  rod,  wood  moulding,  etc.,  up  to  large  items  such  as  cost 
per  mile  of  transmission  line  on  cost  of  a  railroad  crossing,  etc. 

Some  examples  from  such  a  record  are  given  in  the  following: 

(a)  Prices  (at  warehouse)1 

No. 3  Porcelain  insulators $.16  each 

No.  2  T.  B.  W.  P.  wire 2348  per  pound 

Primary  fuse  boxes 3 . 94  each 

30-ft.  6-in.  pole,  rough 7 . 28  each 

etc 

(b)  Labor  formulas. 

Wire  stringing  (single  wire)     Gang  =  (IF  +  17V  +  ICh  +  4L  +  3Gr)  =  G 


Miles  per 
day 

Labor  cost 
per  mile 
without 
loading 

Plus  16.55 
per  cent 
for  loading 

No.  6  Solid                     

2  8 

3575G 

4165G 

No.  0  Stranded 

1  6 

625(7 

728G 

Ground  connections        Gang  =  (17V  + 


23,000- volt  ground  connections 


+  3L  +  IGr)  =  G 

Labor  cost 

each  without     Plus  18.75  per 
loading  cent 

.02075G  .0248G 


Anchors  and  guys    Gang  =  (2/7F  +  2/77V  +  2/7Ch  +  2L  +  IGr)  =  G 


Pole  to  pole  one  %  in ... 
Pole  to  anchor  two  %  in . 
Stub  to  anchor  one  %  in 


Labor  cost       Plus  17.65  per 


each  without 
loading 
.0833G 
. 2833G 
.25G 


cent  for 
loading 
.098£ 
.3333G 
.2941£ 


1  The  figures  given  below  must  not  be  taken  as  representing  current 
prices.     They  are  given  for  example  only. 


16 


ECONOMICS  OF  ELECTRICAL  DISTRIBUTION 


Crossarms     Gang  =  1/3F  +  1/37Y  +  l/3Ch  +  L  +  Gr)  =G 


Labor  cost 

each  without 

loading 


23,000-volt,  single-arm,  36  in 

23,000-volt,  double-arm,  64  in 

23,000-volt,    double-arm,    64    in.     (strain 
insulators,  both  ways)  etc 

(c)  Unit  costs. 


.131G 


.138G 


Plus 

17.65  per  cent 
for  loading 
.047G 
.  154G 

.162G 


Plus 

Plus 

Mate- 
rial 

per 
cent. 

Labor 

per 
cent 

Total 

loading 

loading 

3  No.  6  —  Secondary,  1,000  ft  

$99.00 

(21.15) 

$120.00 

$34.35 

(16.55) 

$40.00 

$160.00 

2  No.  4—  Primary,  1,000  ft  

91.40 

110.80 

25.46 

29.70 

140.50 

(22.64) 

(17.65) 

3M    X    4M    X    92  in.,   six-pin 

.74) 

crossarm  and  hardware  

M) 

1.64 

1.04 

1.22 

2.86 

(21.80) 

No.  3  porcelain  insulator  

.16 

.21 

Labor  included  in  cost 

of  stringing  wire 

(26.08) 

(20.95) 

Pin  —  1%  X  10>£  X  1  in  

.06 

.076 

.10 

.48 

.196 

(16.91) 

(12.16) 

Poles  —  40  ft.  7  in 

16.70 

19.50 

10.26 

12.48 

31.98 

etc. 

COSTS 


17 


(d]  Assemblies. 

15  kva.  S<A  transformer  installation 


Pole  material 

Plus 
load- 
ing, 
per 
cent 

9  —  No.  3  Porcelain  insulators  at  .16  
6  —  Glass  insulators  at  .043 

$  1.44 
.258 

31.80 
31  80 

$     1.90 
34 

15  —  Screw  brackets  at  .22  

3.30 

26.08 

4.16 

2  —  Six-pin  crossarms  at  .74  
2  —  Blocks   at  .48 

1.48 
96 

22.64 
22  64 

1.82 
1  18 

6  Braces  at    1275 

765 

22  64 

94 

10  —  %-in.  bolts  at  .02  
4  %-in   bolts  at   12 

.20 

.48 

22.64 
22  64 

.245 
59 

3  —  Lags  at  038 

114 

22  64 

14 

2  —  Primary  fuse  boxes  at  3.94  
2  —  Secondary  fuse  boxes  at  1.00          .... 

7.88 
2  00 

22.64 
22  64 

9.67 
2  45 

2  —  Lightning  arresters  4  86 

9  72 

22  64 

11  91 

Ground-rod,  cap  and  moulding  
5  —  ib   No   e  wjre                          

1.58 
1.395 

23.79 
21  15 

1.95 
1  69 

6.6  —  Ib.  No.  4  wire  

1.78 

21.15 

2.16 

30  —  ft.^6~m-  galvanized-iron  wire  
Staples,  screws  etc 

.525 
17 

22.64 
22  64 

.63 
21 

Total  
Labor  1/3(1^  +  ITr  +  3L  +  IGr)  =  $13.90 

$34.05 
plus  15.45 

per  cent 

$  41.99 
16.10 

15-kva.  transformer  $185.90  plus   16.91  per 

cent 

$  58.09 
217  00 

Total.. 

$275.09 

Cost  per  1,000  ft.  of  line  with  125-ft.  span. 


Cross- 

Wire, 

Plus 

Plus 

arms, 

Pins 

Insula- 

Wire 

pins, 

cross- 

35-ft. 

one  per 

tors 

insula- 

arms 

poles 

pole 

tor 

Primary 

2  No.  6  

$22  .  80 

$3.14 

$3.36 

$106.80 

$113.30 

$136.10 

$335  22 

2  No.  2  

22.80 

3.14 

3.36 

186.15 

192.65 

215.45 

414.57 

Secondary 

3  No.  4  

22.80 

4.70 

1.37 

210.00 

216.87 

239.67 

438.79 

18  ECONOMICS  OF  ELECTRICAL  DISTRIBUTION 

The  above  are  merely  examples  selected  here  and  there  and  in 
no  way  indicate  the  complete  record.  It  is  easily  seen  that  the 
compiling  of  such  a  record  is  a  matter  of  considerable  labor  and 
investigation.  Once  obtained,  however,  in  this  form  a  revision  is 
a  comparatively  simple  matter. 

While  the  costs  thus  given  are  average  figures,  especially  for 
labor,  they  may  be  applied  in  the  locality  in  which  they  were 
derived  without  great  error  for  estimating  individual  jobs  unless 
some  unusual  field  conditions  indicates  extraordinary  additions 
to  some  part  of  the  cost,  For  economic  studies  of  course  average 
figures  are  usually  desirable. 

ANNUAL  COSTS 

The  determination  of  annual  costs  as  used  in  this  work  is 
naturally  divided  into  two  parts,  that  pertaining  to  the  physical 
property  itself  such  as  construction  and  maintenance  costs  and 
that  pertaining  to  the  load  carried,  i.e.,  cost  of  energy.1 

Annual  Costs  on  Physical  Property. — The  items  to  be  con- 
sidered under  the  first  heading,  i.e.,  annual  costs  on  the  physical 
property,  will  each  be  discussed  briefly.  They  include,  interest, 
taxes,  insurance,  maintenance,  repair  and  depreciation. 

Interest. — Whenever  money  is  invested  in  a  piece  of  property 
a  legitimate  rate  of  interest  must  be  expected  as  part  of  the 
earnings  of  that  property  unless  it  be  run  at  a  loss.  Interest 
must  be  figured  on  the  total  investment  involved  including  all 
material,  labor  and  overhead  costs.  The  rate  at  which  interest 
should  be  charged  may  vary  with  the  problem  under  considera- 
tion. Fundamentally  it  should  be  the  current  rate  of  interest 
on  sound  investments  or  the  rate  at  which  the  company 
could  borrow  money  under  ordinary  conditions.  Sometimes  the 
average  rate  paid  on  total  capitalization  may  be  taken  but  in 
case  the  dividend  on  capital  stock  is  fairly  large,  part  of  it  might 
be  considered  as  a  profit  in  excess  of  a  fair  rate  of  interest.  In 
some  cases  due  to  poor  financial  conditions  or  for  an  emergency 
a  company  might  have  to  pay  a  higher  rate  for  money,  even 
on  bonds,  than  the  market  rate.  All  such  factors  should  be 
considered  in  determining  the  interest  rate  to  be  used. 

Taxes. — Taxes  are  an  ever-present  charge  on  any  property  and 
usually  the  definite  percentage  may  be  easily  determined  from 
the  company's  accounting  records. 

1  For  determination  of  " Energy  Cost"  see  Chapter  IV. 


COSTS  19 

Insurance. — Insurance  against  loss  by  fire  is  the  most  common 
form  but  on  some  classes  of  property  insurance  against  theft, 
storm,  etc.  is  also  carried.  In  any  problem  the  kind  and  amount 
of  insurance  chargeable  to  each  class  of  property  should  be  investi- 
gated. In  many  cases  no  " Insurance"  charge  is  necessary. 

Maintenance  and  Repair. — Maintenance  and  repair  will  be 
different  for  each  unit  considered.  No  two  transformers  for 
example  will  require  the  same  amount  of  attention  during  their 
life  even  though  similarly  located:  breaks  in  a  line  can  rarely  be 
anticipated,  etc.  A  large  part  of  maintenance  is  occasioned  by 
imperfection  in  material.  Maintenance  and  repairs  due  to  such 
causes  on  any  individual  piece  of  equipment  cannot  be  foreseen. 
In  such  cases  average  figures  only  can  be  obtained  from  actual 
experience  over  a  number  of  years.  Other  items  of  maintenance 
such  as  inspection,  testing,  etc.,  can  be  quite  definitely  determined 
from  payroll  and  .time  reports. 

Depreciation— -No  detailed  discussion  of  depreciation  can  be 
here  included.  Strictly  speaking,  depreciation  is  the  percentage 
by  which  a  piece  of  property  is  reduced  in  value  each  year  of  its 
life  (by  value  is  not  meant  necessarily  selling  price).  From  an 
accounting  point  of  view,  on  the  other  hand,  a  certain  amount 
must  be  set  aside  each  year  to  replace  the  property  when  worn 
out.  Sometimes  the  usefulness  of  the  property  in  service  is  con- 
sidered as  a  measure  of  its  value.  All  these  different  viewpoints 
give  rise  to  different  methods  of  figuring  depreciation,  and  a 
discussion  of  these  may  be  found  in  other  works.  For  the  pur- 
pose of  this  work  however  what  is  known  as  the  straight-line 
method  is  probably  the  simplest  and  most  satisfactory.  This 
method  considers  a  piece  of  property  to  have  given  service  for 
the  years  of  its  life  for  a  definite  total  cost  which  may  be  equally 
divided  between  the  years.  The  cost  is  obtained  from  the  total 
first  cost  of  the  property  in  place  including  material,  labor  and 
overhead  charges,  less  the  salvage  value  at  the  end  of  its  life,  plus 
the  labor  cost  necessary  to  salvage  it.  This  cost  is  divided  by  the 
estimated  number  of  years  of  life  and  the  percentage  of  deprecia- 
tion taken  as  the  percentage  of  the  total  first  cost  thus  obtained. 
Naturally  this  will  vary  with  different  classes  of  property.  Some 
will  have  little  if  any  salvage  value  such  as  crossarms,  for 
example.  A  pole,  when  rotted  at  the  base,  on  the  other  hand  can 
be  sawed  off  and  used  again  either  as  a  shorter  pole  or  a  stub. 
Bare  copper  wire  will  have  practically  no  physical  depreciation, 


20  ECONOMICS  OF  ELECTRICAL  DISTRIBUTION 

the  cost  of  stringing  and  removing  together  with  tie  wires  and 
other  incidentals  making  up  the  depreciation. 

The  years  of  life  to  use  for  any  unit  may  depend  on  other 
things  than  its  own  life.  For  a  transmission  line,  for  example,  it 
is  probable  that  no  definite  limit  can  be  fixed  at  which  the  whole 
line  must  be  replaced.  Poles  and  crossarms  will  be  replaced 
from  time  to  time  when  necessary  as  long  as  the  line  is  in  service. 
In  such  a  case  it  is  probably  simplest  to  assume  a  definite  life  for 
the  whole  line,  possibly  the  assumed  life  of  a  pole,  and  figure 
depreciation  on  all  units,  wire,  insulators,  etc.  on  that  basis. 
Thus  any  class  of  property  may  have  a  different  percentage  of 
depreciation  depending  on  where  it  is  used. 

A  simple  example  of  a  computation  for  depreciation  on 
insulated  wire  would  be  as  follows : 

Assuming  new  wire  at  30  cts.,  scrap  copper  at  20  cts.  per  pound. 
Cost  of  No.  0  wire  per  1,000  ft.-420  Ib.  at  30  cents  =  $126.00 

Labor  of  stringing  (assumed) 10 .  00 

Labor  of  salvaging  (including  burning  off  insulation)       10.00 


Total $146.00 

Salvage  value  319  Ib.  at  20  cts.  per  pound 63 . 80 

Net  Cost $82.20 

If  the  assumed  life  of  insulation  is  15  years 

82.20  5.47 

=  $5 . 47  per  year  —  =  4  per  cent  per  year. 

15  136 . 00 

The  matter  of  obsolescence  which  is  sometimes  considered  as 
a  separate  figure  may,  for  this  work,  be  considered  as  a  part  of 
depreciation.  Where  it  is  anticipated  that  materials  will  become 
obsolete  and  require  replacement  before  worn  out  on  account  of 
improvement  in  design,  the  assumed  life  and  salvage  value  used 
in  computing  depreciation  should  be  adjusted  accordingly. 

As  an  example  of  the  above,  the  total  percentage  of  annual 
charge  on  a  pole  line  may  be  taken  as  15  per  cent,  made  up  as 
follows : 

PER  CENT 

Interest 7 

Taxes 2 

Insurance 0 

Depreciation 6 

Many  special  problems  arise  in  figuring  annual  cost.  One 
which  also  includes  the  idea  of  obsolescence  is  encountered  when 


COSTS  21 

the  replacement  of  a  serviceable  line  with  one  of  larger  capacity 
is  being  considered.  The  value  remaining  in  the  old  line  must  be 
included  in  the  computations  and  the  total  investment  repre- 
sented in  the  new  line  must  include,  in  addition  to  the  cost  to 
build  it,  the  present  value  of  the  labor  necessary  to  build  the  old 
line,  i.e.,  first  cost  for  labor  less  depreciation  for  years  of  age  and 
the  labor  cost  necessary  to  dismantle  the  old  line.  Such  prob- 
lems will  be  discussed  in  more  detail  later. 


CHAPTER  IV 
ENERGY  COST 

PRINCIPLES  AND  METHODS  INVOLVED  IN  THE  DETERMINATION  OF 
THE  COST  OF  ENERGY  AND  OF  ENERGY  LOSSES 

Since  any  economic  study  of  transmission  or  distribution  is, 
fundamentally,  a  consideration  of  the  cost  of  energy  or  energy 
losses  as  compared  with  other  costs  (in  general,  fixed  charges 
increase  as  energy  loss  decreases),  it  is  of  the  utmost  importance 
that  the  cost  of  energy  be  accurately  determined.  The  value 
assigned  to  the  unit  cost  of  energy  may  be  the  deciding  factor  in  a 
problem.  This  value,  moreover,  may  vary  considerably  according 
to  the  assumptions,  methods  and  precision  employed  in  its  calcu- 
lation. It  is  evident  that  the  cost  per  kilowatt-hour  of  the  energy 
used  by  a  5-h.p.  motor  running  1  hr.  per  day,  25  miles  north  of  the 
generating  station  may  be  quite  different  from  the  unit  cost  for 
residence-lighting  load,  5  miles  west  of  the  station  and  both  of 
these  may  be  far  from  the  average  unit  cost  over  the  whole 
system.  The  question  of  the  determination  of  energy  cost 
offers  an  extensive  field  for  study  and  one  that  generally  has  been 
only  touched  upon.  This  chapter  will  give  some  of  the  funda- 
mental principles  involved,  a  few  methods  of  attacking  the  general 
problem  and  suggestions  as  to  conditions  governing  variations 
in  the  general  cost  as  applied  to  particular  uses. 

In  studying  energy  cost,  the  fundamental  difference  between 
the  cost  of  energy  for  rate  making  purposes  and  the  cost  of  energy 
for  economical  study  must  be  recognized.  '  In  determining  the 
cost  of  energy  for  the  purpose  of  adopting  a  rate  scale,  it  may  be 
sufficient  to  consider  the  system  as  a  whole  and  determine  the 
average  cost  per  unit  for  each  of  a  few  general  classes  of  load 
which  have  markedly  different  characteristics.  The  chief  point 
to  be  kept  in  mind  is  the  amount  which  the  customer  pays,  and 
that  this,  on  an  average  should  at  least  equal  the  expense  of  the 
company,  plus  a  reasonable  profit.  Usually  about  the  same 
rate  must  be  applied  to  similar  customers  within  a  reasonable 
area,  unless  there  is  a  marked  difference  in  the  individual  cost  of 
serving  each.  There  are  also  other  factors  than  actual  produc- 
tion and  distribution  cost  which  must  be  considered  in  rate  making, 
such  as  public  opinion,  regulations  and  franchise  agreements,  pre- 
vious practice,  competition,  etc.  On  the  other  hand,  in  making  an 

22 


ENERGY  COST  23 

economic  study,  we  are  interested  in  the  actual  amount  which  the 
energy  delivered  to  the  point  under  consideration  is  costing  the 
company  and  hence,  how  much  money,  if  any,  can  be  saved 
by  reducing  energy  losses.  If  energy  costs  more  per  unit  5  miles 
from  the  station  than  it  does  1  mile,  for  example,  the  amount 
saved  will  be  correspondingly  more  important.  For  this  pur- 
pose, then,  it  would  seem  desirable  to  investigate  the  cost  of 
energy  as  fully  as  is  warranted  by  the  amount  and  accuracy  of 
the  information  available  on  costs  of  construction  and  operation 
and  on  loads  carried. 

Even  with  fairly  complete  data  at  hand  it  is  a  difficult 
matter  to  determine  definite  figures  for  energy  costs.  The  cost  is 
affected  by  many  quantities  of  a  variable  nature  and  these  limit 
the  extent  to  which  it  is  practicable  to  carry  the  study.  At  any 
given  point  the  cost  of  energy  is  largely  dependent  on  the  fixed 
charges  and  operating  expenses  of  the  central  station  and  of  the 
distribution  system  between  the  station  and  that  point.  It  is 
also  affected  by  the  size  and  characteristics  of  the  load  at  the 
point  in  question  in  relation  to  all  the  other  loads  on  the  system. 
Hence,  strictly  speaking,  energy  cost  may  be  conceived  as  having 
a  different  value  at  every  point  on  the  system  and  for  every 
different  load  at  any  given  point.  Loads  of  the  same  type  may 
show  a  different  unit  cost  according  to  their  size  and  the  unit  cost 
may  vary  at  different  times  during  the  day  or  even  for  different 
parts  of  the  same  load.  The  length  to  which  the  determination 
of  cost  of  energy  might  be  carried  is  almost  infinite.  In  practice 
it  will  depend  not  only  on  the  accuracy  of  the  data  available  but 
also  on  how  this  cost  is  to  be  used. 

Classification  of  Costs  of  a  Central-station  System. — The 
various  items  entering  into  the  total  annual  cost  of  a  central 
station  may  be  classified  as  follows: 

1.  Costs  dependent  on  the  number  of  customers. 

2.  Costs  dependent  on  the  peak  load  carried  or  the  demand. 

3.  Costs  dependent  on  the  total  output  in    kilowatt-hours 
during  the  year. 

This  method  of  classification  which  was  suggested  by  Dr. 
John  Hopkinson  in  England  in  1892  and  was  later  enlarged  upon 
by  H.  L.  Doherty  and  other  writers,  is  quite  generally  accepted 
as  sufficient  for  ordinary  purposes.  It  must  not  be  assumed 
that  all  expenses  come  strictly  under  one  of  these  three  classifi- 
cations. There  are  a  great  number  of  other  minor  divisions  which 


24  ECONOMICS  OF  ELECTRICAL  DISTRIBUTION 

might  be  made.  For  example,  there  are  certain  operating 
expenses  at  the  station  which  are  dependent  on  the  efficiency 
and  size  of  the  machines  and  the  relation  of  the  load  at  any  time 
to  the  capacity  of  the  generators  in  use  and  to  the  method  of 
operating  the  station  and  the  system.  It  does  not  appear 
practical  however  to  attempt  to  include  all  such  classifications. 
The  above  three  are  the  ones  of  most  importance  and  other 
costs  can  be  included  in  one  of  them  without  great  error. 

Consumer's  Cost. — The  first  division,  consumer's  cost,  usually 
need  not  be  considered  in  determining  energy  cost  for  economic 
study,  as  it  is  independent  of  the  amount  of  load  carried.  This 
cost  occurs  beyond  the  limits  of  the  lines  and  need  be  added 
only  when  determining  the  charge  to  the  customer.  Care  must 
be  taken  however  that  charges  actually  belonging  to  this  classi- 
fication are  not  included  under  either  of  the  other  heads.  Here 
rightfully  belong  the  greater  part  of  charges  for  general-office 
expense,  sales  expense,  metering,  billing,  collection,  etc.,  part  of 
cost  of  service  wires  and  a  percentage  of  various  other  charges 
according  to  local  conditions. 

Demand  Cost  and  Output  Cost. — The  other  two  cost  divisions, 
demand  cost  and  output  cost,  must  include  all  charges  which  are 
dependent  on  the  load  carried.  The  demand  cost  is  that  part 
of  the  total  cost  which  is  caused  by  the  fact  that  the  system 
is  built  and  operated  to  care  for  a  certain  maximum  load.  If 
this  peak  load  is  100,000  kw.  the  station  must  have  a  capacity  of 
at  least  100,000  kw.,  with  a  reasonable  amount  of  reserve,  regard- 
less of  the  fact  that  that  capacity  may  be  reached  for  only  a 
short  time  each  day.  The  output  cost  or  kilowatt-hour  cost  on 
the  other  hand  is  that  part  of  the  total  cost  which  is  occasioned 
by  the  total  output  in  kilowatt-hours  regardless  of  the  rate  of 
that  output. 

Some  items  of  the  total  annual  cost  clearly  belong  only  to 
demand  cost  while  others  which  might  seem  to  depend  only  on 
output  have  some  percentage  of  demand  cost  included.  In 
the  first  group  come  interest,  taxes,  insurance,  depreciation,  etc. 
on  the  generating-station  building,  interest,  taxes,  etc.  on  boilers, 
turbines,  generators  and  other  equipment  and  a  large  part  of 
their  depreciation  and  maintenance.  Also  fixed  charges  on  lines 
and  substation  equipment  are  here  included.  Under  the  second 
group  comes  part  of  the  cost  of  coal,  oil  and  other  expendable 
materials,  part  of  the  labor  of  operating,  also  part  of  the  energy 


ENERGY  COST  25 

losses  on  lines  and  transformers  due  to  the  fact  that  they  are 
kept  energized  at  all  times.  The  apportioning  of  such  costs  is  a 
matter  for  considerable  study.  The  method  of  operation  may 
effect  the  amount  chargeable  to  demand  in  certain  cases.  Large 
machines  are  less  efficient  at  small  loads.  Hence  if  large  units 
are  employed  and  are  run  far  below  their  most  economical  load 
for  the  greater  part  of  the  day,  the  extra  expense  due  to  decreased 
efficiency  may  be  charged  to  demand.  In  a  large,  efficiently 
operated  station  this  condition  would  not  occur  to  any  great 
extent,  but  it  suggests  some  of  the  items  which  must  be 
considered.  In  fact  there  must  be  included  in  demand  cost  all 
charges  directly  or  indirectly  occasioned  by  the  total  capacity  of 
the  system.  The  remaining  annual  costs,  aside  from  the  con- 
sumer cost  before  mentioned,  may  be  considered  as  dependent  on 
the  output.  These  comprise  the  kilowatt-hour  cost. 

Methods  of  Making  Classification. — The  actual  division  of 
the  total  annual  cost  on  any  part  of  the  system  into  three  classifi- 
cations will  depend  largely  on  local  conditions.  The  relative  per- 
centages will  probably  be  different  for  each  company.  Several 
general  methods  of  attacking  the  problem  are  in  use.  In  the 
generating  plant,  for  example,  an  empirical  analysis  can  be  made 
of  each  item  of  cost,  such  as  charges  on  building,  on  steam  equip- 
ment, on  electrical  equipment,  fuel,  labor,  etc.  The  proportion 
belonging  to  each  classification  may  be  estimated  from  known 
conditions,  keeping  in  mind  the  general  definitions  of  demand 
cost,  kilowatt-hour  cost  and  consumer  cost.  One  method  of 
separating  demand  and  kilowatt-hour  charges  on  fuel,  lubricants 
and  such  items  is  by  comparing  costs  under  a  period  of  light  load 
and  one  of  heavy  load — a  month  at  different  seasons  of  the  year. 
For.  example, 

if  CD  =  demand  cost,  per  unit  demand 

€/  =  kilowatt-hour  cost, 

D  =  station  demand, 

FI  =  kilowatt-hours  at  low  period, 

Fz  =  kilowatt-hours  at  high  period, 

Ci  =  total  cost  at  low  period, 

C2  =  total  cost  at  high  period, 

Ci  =  CDD  +  cfFl 

C2  =  CDD  +  C}FZ 

C2  -  Cl  Cl-C/Fl 


26  ECONOMICS  OF  ELECTRICAL  DISTRIBUTION 

This  should  give  satisfactory  results  if  there  were  enough 
difference  between  the  loads  at  the  two  periods  to  give  a  good 
comparison  of  cost. 

Another  somewhat  similar  method,1  that  may  be  applied  to 
items  such  as  total  station  or  system  costs,  which  involve  all  three 
classifications,  considers  the  total  cost  over  three  given  periods, 
three  years  for  example  or  three  different  months.  Using  similar 
symbols  to  the  above  with 

Cc  =  customers  cost, 

N  =  number  of  customers, 

Ci  =  CnD,  +  C,F!  +  CCN1 


If  these  are  solved  simultaneously  the  values  of  CD,  C/  and 
Cc  can  be  determined. 

An  example  of  a  typical  division  of  costs  between  demand  and 
kilowatt-hour  of  some  of  the  items  for  the  generating  station  is  as 
follows  : 

Demand,  Kilowatt- 

per  cent  hour,  per  cent 
Superintendence  ..................             100 

Wages  ...........................               90  10 

Fuel  .............................              25  75 

Lubricants  .......................              25  75 

Station  supplies,  etc  ...............  100 

These  general  methods  may  be  adapted  to  other  parts  of  the 
system,  transmission  lines,  substations,  etc.,  as  well  as  to  the 
generating  plant.  In  analyzing  the  cost  of  a  hydro-electric 
plant,  the  available  supply  of  water  is  an  important  factor  in  the 
consideration.  It  is  evident  that  the  kilowatt-hour  cost  will 
depend  considerably  on  whether  the  supply  is  unlimited  or 
whether  storage  is  resorted  to  for  regulating  the  flow. 

It  appears  then  that  the  first  step  in  the  study  of  energy  cost 
is  the  determination  as  accurately  as  possible  of  the  total  annual 
costs  on  each  subdivision  of  the  system  and  the  proportion  of 
these  costs,  in  each  case,  belonging  to  demand  and  to  output. 
Naturally  the  more  detailed  the  accounting  record  on  various 
parts  of  the  system,  the  easier  the  determination  of  these  costs 

1  This  method  is  more  fully  explained  in  ''Central  Station  Rates  in  Theory 
and  Practice,"  by  H.  E.  EISENMENGER,  Electrical  Review,  Vol.  75,  Aug.  23, 
1919,  p.  305. 


ENERGY  COST  27 

will  be.  If  an  entirely  new  system  is  being  considered,  the 
various  quantities  can  be  only  estimated  from  data  of  other 
similar  systems  and  present  prices  on  construction  materials, 
equipment,  etc.  For  the  purpose  under  consideration,  if  the 
study  of  energy  cost  is  to  be  carried  to  any  degree  of  exactness, 
it  is  probably  better  to  prepare  separate  figures  on  each  of  the 
three  classifications  of  costs  as  related  to  different  parts  of  the 
system,  generating  station,  various  substations,  underground- 
cable  lines,  overhead  power  lines,  feeders,  etc.  Just  how  much 
of  such  detail  is  necessary  will  be  determined  by  the  method 
which  is  to  be  used  in  apportioning  the  costs  among  the  various 
classes  of  loads  and  localities. 

Apportioning  of  Demand  and  Output  Costs  to  Various  Types 
of  Loads. — When  the  proportions  of  the  total  cost  chargeable  to 
demand  and  to  output  have  been  determined,  there  arises  the 
problem  of  finding  what  part  of  that  demand  cost  or  of  that 
kilowatt-hour  cost  belongs  to  any  particular  load  or  type  of 
load.  The  Kilowatt-hour  cost  may  be  simply  disposed  of  for  the 
present  by  considering  that  the  kilowatt-hour  charge  at  any 
point  in  the  system  is  equal  to  the  sum  of  the  kilowatt-hour 
costs  incurred  on  all  parts  of  the  system  from  that  point  back  to 
the  generating  station.  Considerable  study  may  be  involved  in 
determining  the  kilowatt-hour  costs  on  such  parts  as  trans- 
mission lines  or  power  lines,  since  a  great  part  of  the  cost  depends 
on  the  energy  losses  and  the  cost  of  these  losses  in  turn  includes 
the  demand  and  kilowatt-hour  charges  up  to  that  point.  The 
theory  is  not  complex  however.  Modification  of  this  method  for 
special  purposes  will  be  explained  later.  The  apportioning  of 
the  demand  cost  however  presents  a  more  difficult  question. 

Demand  Cost. — If  all  loads  had  similar  characteristics,  i.e., 
similar  load  curves,  with  the  peak  coming  at  the  same  time  it  is 
evident  that  the  demand  charge  for  each  would  be  simply  pro- 
portional to  its  peak  load.  This  assumption  is  sometimes  used 
in  figuring  energy  cost  but  is  obviously  not  correct  except  in 
cases  where  all  loads  are  similar  or  nearly  so,  such  as  for  a  plant 
serving  residence  lighting  only.  On  this  assumption,  the 
demand  charge  per  kilowatt  may  be  reduced  to  a  figure  per 
kilowatt-hour,  inversely  proportional  to  the  load  factor,  and 
this  added  to  the  kilowatt-hour  charge  determines  the  total 
cost  of  energy.  Where  both  lighting  and  commercial  power 
or  other  loads  are  carried,  however,  it  is  evident  that  the  demand 


28 


ECONOMICS  OF  ELECTRICAL  DISTRIBUTION 


responsibility  of  each  cannot  be  accurately  obtained  in  this  way 
as  the  peaks  come  at  different  times  and  the  loads  show  different 
characteristics  throughout  the  day.  For  example  the  power 
peak  might  come  at  the  same  time  as  the  station  peak  at  perhaps 
2  P.M.,  whereas  the  lighting  peak  comes  at  8  P.M.,  the  lighting 
load  at  2  P.  M.  being  only  30  per  cent  of  its  peak. 

Again  the  assumption  might  be  made  that  the  demand  charge 
of  any  load  is  proportional  to  its  demand  at  time  of  station  peak. 
This  might  sound  reasonable  inasmuch  as  the  demand  cost  of 
the  station  is  figured  on  the  basis  of  its  peak  load.  The  demand 
responsibility  for  any  customer  is  sometimes  computed  on  this 
basis  by  multiplying  his  total  connected  load  by  his  demand  fac- 
tor to  get  his  individual  demand,  dividing  this  in  turn  by  the 
diversity  factor  on  the  line  for  his  proportion  of  the  line  demand, 
this  by  the  diversity  factor  of  the  substation  and  so  on  back  to 
the  generating  station,  thus  determining  that  customer's  propor- 
tion of  the  station  peak.  Of  course  a  careful  determination  of 
diversity  and  demand  factors  is  necessary  for  this.  Here  again 
the  variable  characteristics  of  the  different  loads  make  this 
assumption  erroneous  except  in  special  cases. 

Take  for  a  simple  example  a  small  station  of  1,000  kw.  serving 
two  customers  A  and  B.  A  takes  1,000  kw.  for  6  hr.  each  day. 
B  takes  600  kw.  for  the  remaining  18  hrs.  but  does  not  overlap  A. 
On  the  assumption  of  demand  proportional  to  individual  peak 
load,  A's  cost  would  be  x%6  and  B's  %Q  of  the  total  demand 
cost.  On  the  assumption  of  demand  proportional  to  load  at 
time  of  station  peak,  A's  proportion  would  be  the  whole  station 
demand  and  B's  nothing  (see  Fig.  1). 


FIG.  1. 


If,  however,  the  station  may  be  considered  as  consisting  of  two 
parts,  one  of  a  capacity  of  600  kw.  and  one  of  400  kw.,  it  will  be 
readily  seen  that  the  600  may  be  assumed  to  operate  6  hr.  for  A 
and  18  for  B  while  the  400  operates  only  6  hr.  for  A  (see  Fig.  2). 


In   this   case   then   A's  cost  would  be 


400 
1,000 


A  v  60° 

24  X  1,000 


ENERGY  COST 


29 


=          °f 


X 


of  the  total.  This  theory  can  be  extended  to  cover  any  number 
of  loads  with  variously  shaped  curves.  It  may  be  stated  in 
general  that  the  cost  of  each  unit  (kilowatt)  of  the  total  demand 
should  be  divided  in  accordance  with  the  length  of  time  or  number 
of  hours  it  is  in  use,  to  obtain  an  accurate  apportioning  of  demand 
cost.  A  full  explanation  of  this  theory  with  examples  of  its 
application  may  be  found  in  "Central  Station  Rates  in  Theory 


FIG.  2. 

and  Practice,"  by  H.  E.  Eisenmenger,  Electrical  Review,  Vol.  75, 
Aug.  2  and  9,  1919.  Ordinarily  sufficient  data  in  regard  to  the 
various  load  curves  may  not  be  available  or  the  degree  of  accuracy 
desired  in  the  result  will  not  warrant  an  extensive  analysis  on  this 
basis.  There  is  no  doubt  that  theoretically  it  will  give  a  more 
accurate  distribution  of  the  demand  cost  for  most  purposes  than 
either  of  the  other  methods  mentioned  and  the  principles  involved 
may  often  be  used  to  advantage,  even  in  a  more  approximate 
determination.  For  some  special  uses,  as  will  be  explained  later, 
the  second  method  given  above  is  preferable. 

General  Method  for  Determining  Demand  Cost  at  any  Point. 

In  order  to  determine  the  demand  cost  at  any  point  on  the 
system  other  than  the  generating  station  the  following  general 
steps  should  be  followed  (see  Fig.  3) : 

1.  Apportion  the  generating-station  demand  cost  among  the 
various  substations  by  one  of  the  above  methods,  in  accordance 
with  their  loads. 

2.  For  any  substation,  add  to  its  portion  of  the  generating 
station  cost  the  demand  costs  on  its  transmission  cables  and  on 
the  substation  itself. 

3.  The  total  demand  cost  for  the  station  can  now  be  distributed 
among  the  various  lines  feeding  out  from  it  and  by  repeating  the 
same  method  the  cost  at  any  point  or  for  any  customer  may  be 
determined.     If  more  general  figures  only  are  desired  the  total 
demand  cost  for  the  station  may  be  divided  among  various  classes 


30 


ECONOMICS  OF  ELECTRICAL  DISTRIBUTION 


of  load  such  as,  lighting,  commercial  power,  street  lighting,  street 
railway,  etc.  By  assuming  average  figures  on  line  costs,  the  cost 
at  any  distance  from  the  station  for  each  class  of  load  can  be 
determined.  The  analysis  may  be  extended  in  a  similar  manner 


GENERATING     STATION 

n          jr    j.       Distributed  accordi'nq 
Demand  Cost       to  load  curves  of 

Djvided  equally  among 
Operating  Cost   kilowatt-hours 
of  output 


•TRANSMISSION  LINES 
AND  CABLES 


SUBSTATION 

Demand  Cost  on 

Substation 
+  Demand  Cost  on 

Transmission 
+  Proportion  of 

Demand^oston 

Generating  Sta. 


Distributed  accord- 
ing to  load  curves 
on  outgoing  loads 


Operating  Cost  on 
Substati'on 

ston 
ansmission 


4- Operating  Co« 
Trans  mi'ssio' 


Divided  equally 
among  kilowatt- 
hours  of  output 
•^-Generating  sta- 
tion kilowatt-hour 
cost 


Demand  Cost  on  I  Mile 

of  Power  Lme 
+  Proportion  of  substation 

Demand  Cost 


Operating  Cpst  on  I  Mile 
of  Power  Line  divided 
equally  among  kilowatt 
hours  of  output. 

•*•  Kilowatt  hour  cost  at 
substation. 


Proportion  of  substation 

Demand  Cost 
+  Demand  Cost  on /^  mile 


Operating  Costpn"I^Tmile 

of  Distribution, 

divided  equally  among 

kilowatt  hours  of 

output 
+  kilowatt -hour  cost 

at  substation 


FIG.  3. — Diagram  indicating  general  method  of  studying  energy  cost  at  various 

points  on  a  system. 

to  cover  suburban  transmission  lines,  substations  and  distribution 
even  where  several  substations  intervene. 

Any  such  analysis  should  give  a  fairly  accurate  demand  charge 
at  any  point.  It  requires  however  quite  a  large  amount  of 
accurate  data  for  its  accomplishment.  Detailed  annual  costs 


ENERGY  COST  31 

on  each  subdivision  of  the  system  must  be  known.  Also  the  load 
curves  throughout  the  year,  for  each  load  considered  such  as 
substations,  power  lines,  etc.,  must  be  studied  to  obtain  one  or 
more  characteristic  curves  for  each  as  a  basis  of  analysis.  Where 
a  load  shows  an  appreciable  seasonal  variation  several  curves 
may  be  necessary  as  for  high  load,  low  load  and  average  load. 
In  many  cases  it  will  be  found  practicable  to  determine  a  general 
characteristic  curve  for  each  type  of  load,  such  as  residence  light- 
ing, street  lighting, .  street  railway,  large  power,  small  power, 
etc.,  and  refer  all  loads  of  that  type  to  it,  assuming  that  the  curve 
will  be  always  proportional  to  the  peak  load.  The  collection, 
classification  and  proper  selection  of  data  will  be  found  to  be  a 
large  part  of  the  whole  problem  of  determination  of  energy  cost 
and  requires  the  application  of  a  high  degree  of  engineering 
knowledge  and  judgment. 

In  order  to  make  such  a  detailed  analysis  as  proposed  above  the 
following  information  should  be  available. 

1.  Cost  of  Generating  Station. 

(a)  First  cost  and  present  value  of  various  parts  in  sufficient  detail  so 
that  demand,  energy  and  customer's  costs  can  be  separated. 

(6)  The  proper  percentages  of  interest,  taxes,  depreciation,  etc., 
chargeable  to  each  part. 

(c)  Operating  costs  in  some  detail. 

(d)  Determination  of  proper  percentage  of  each  item  of  fixed  and 
operating  charges  belonging  to  each  of  the  three  divisions  of  cost. 

2.  Cost  of  Transmission-line  Cables. 

(a)  First  cost  and  present  value  of  cable  lines  to  each  substation  with 

proportional  cost  of  tie  lines  between  stations. 
(6)  As  above. 

(c)  Such  operating  cost  as  may  be  chargeable  to  cables,  repairs,  etc. 

(d)  As  above. 

3.  Cost  of  Substations. 

(a)  First  cost  and  present  value  of  each  substation  in  sufficient  detail. 
(6,  c,  d)As  above. 

4.  Cost  of  Overhead  Lines. 

(a)  First  cost  and  present  value  of  average  line  of  each  class,  trans- 
mission, power  line,  circuit,  direct-current  feeder,  railway  feeder,  and 
secondary  distribution  including  transformers,  per  unit  length. 

(6,  c,  d)  As  above. 

5.  Characteristic  Curve  for  Generating  Station. 

(May  be  taken  for  seasons  or  months  instead  of  for  year.) 

6.  Characteristic  Curves  for  each  Substation. 

(Corresponding  with  generating-station  curves.) 

7.  Characteristic  Curves  for  Each  Class  of  Load  out  of  Each  Substation. 

(Or  for  each  line.) 


32  ECONOMICS  OF  ELECTRICAL  DISTRIBUTION 

Total  Charge  Per  Kilowatt-hour. — All  the  formulas  which  will 
be  developed  later  include  the  cost  of  energy  as  one  total  charge 
per  kilowatt-hour.  The  computations  might  be  made  with  the 
demand  charge  and  the  kilowatt-hour  charge  as  two  separate 
quantities,  but  it  is  found  more  convenient  to  use  a  single 
charge  per  kilowatt-hour.  If  the  above  analysis  is  carried  out 
in  detail  the  demand  charge  for  energy  will  be  determined  as  a 
different  amount  for  each  type  of  load  and  for  any  distance 
from  each  substation.  The  kilowatt-hour,  charge  will  be  the 
same  for  all  loads  in  any  one  locality.  Since  the  average  load 
factor  for  any  type  of  load  considered  may  be  determined,  the 
demand  charge  can  be  reduced  thereby  to  a  charge  per  kilowatt- 
hour,  since  kilowatt  hours  =  kilowatts  X  load  factor  X  24  X  365 
per  year.  This  added  to  the  output  or  kilowatt-hour  charge  will 
give  the  total  charge  per  kilowatt-hour  for  that  class  of  load  at 
that  point. 

Further  Variations  in  Energy  Cost. — Up  to  this  point  there  has 
been  considered,  for  any  type  of  load  and  locality,  only  the 
average  unit  cost  of  the  total  energy  delivered,  without  attempt- 
ing to  differentiate  between  costs  for  large  loads  and  small  loads 
of  the  same  type,  high  power  factors  and  low  power  factors,  losses 
and  used  power,  etc.  There  will  now  be  indicated  some  of  the 
possible  variations  in  this  cost  and  methods  of  studying  them 
will  be  suggested. 

It  may  be  shown  that  all  units  of  energy  even  of  the  same  class 
and  locality  do  not  cost  the  same.  For  example,  energy  lost  has 
a  higher  unit  cost  than  energy  used.  Again,  not  only  does  the 
total  kilovolt-amperes  of  any  load  increase  as  power  factor 
decreases  but  that  increase  costs  more  per  unit  than  the  average 
cost  if  a  100  per  cent  power  factor  were  obtained.  Further, 
under  some  conditions  an  increase  in  a  load  will  cost  more  per 
unit  than  the  former  average. 

The  underlying  theory  involved  may  be  readily  seen  if  we 
consider  the  cost  of  lost  or  waste  energy  for  example.  Most 
substations  are  regulated  in  some  way,  and  if  the  energy  losses 
between  the  substation  and  the  customer  could  be  reduced  the 
current  in  the  transmission  cable  to  the  substation  would  be  also 
reduced.  Since  losses  are  proportional  to  the  square  of  the 
current,  the  cable  losses  eliminated  by  this  reduction  of  the  load 
on  the  cable  would  be  proportional  to  the  difference  in  the  squares 
of  the  currents  before  and  after  reduction.  Hence  it  will  be 


ENERGY  COST  33 

seen  that  reduction  in  loss  per  unit  reduction  in  load  is  greater  than 
the  average  loss  per  unit  of  the  total  load,  after  reduction.  In 
other  words,  the  loss  in  the  cable  due  to  the  upper  part  of  the 
load,  which  may  be  eliminated  and  hence  may  be  considered 
waste,  is  greater  per  unit  than  the  loss  due  to  useful  load.  Since 
this  energy  must  be  supplied  at  the  generating  plant,  the  energy 
generated  will  be  more  per  unit  for  this  waste  energy  than  for  the 
remainder  of  the  load  or  useful  load  and  the  cost  will  hence  be 
more.  This  may  be  shown  mathematically  as  follows: 

Suppose  a  substation  B  is  supplied  through  cables  from  a 
generation  station  A  — 

(A)  -  :  -  :  -  _(£)• 

W  =  load  in  watts  at  B, 

P  =  percentage  of  W  lost  beyond  B,  which  may  be  considered 
waste  power  which  might  be  conserved  in  some  way, 
p  =  P/100. 
E  =  Voltage  at  B, 
R  =  Resistance  of  cable. 

For  simplicity  assume  single-phase  cable,  unity  power  factor. 
Then 

f  /w\  2 
PR  losses  in  AB  due  to  total  load  W  =  2  f        R, 


I2R  losses  in  AB  due  to  power  load  only  =  2/  —  — 

I2R  losses  in  AB  due  to  losses  beybnd  B(=pW)  = 

r~/W\  2  /W\  2  /W\  2 

^(r)  *-(?)*+•(?) 

or  expressed  in  percentage  of  total  loss  in  AB 

=  100(2p  -  p2)  =  2P  -          per  cent  (1) 


If  the  loss  or  waste  power  at  B  is  a  comparatively  small  per- 
centage of  W  it  will  be  seen  that  the  losses  in  AB  due  to  that 
waste  are  nearly  twice  that  percentage  of  the  total  losses.  (Ten 
per  cent  waste  at  B  means  that  19  per  cent  of  losses  in  A  B  are  due 
to  that  waste.) 

If  Q  =  percentage  of  W  lost  between  A  and  B, 
QW 


.   =  total  loss  in  cable, 


QW 
100 


(P2  \ 
2P  —  -^JQ)  Moo  —  loss  in  cable  due  to  waste  beyond  B. 


34  ECONOMICS  OF  ELECTRICAL  DISTRIBUTION 

PW 

But  waste  beyond  B  =  -77^- 

100 

Hence,  the  loss  in  the  cable  due  to  waste  beyond  B  per  unit 
waste  = 

QTF 
Too 


/        P  \ 

=«/ioo(2-iS)) 


100 


Since  the  average  total  loss  in  cable  per  unit  total  load  at 
B  =         .     If  cost  of  energy  per  kilowatt  at  A  =  Ce 


Average  cost  at  A  of  total  energy  at  B  =  M  +  ~^\  Ce        (3) 
Average  cost  at  A  of  energy  waste  beyond  B 


In  other  words  a  loss  of  Q  per  cent  in  line  A  B  increases  the  aver- 
age cost  of  total  energy  delivered  at  B  by  Q  per  cent.  But  if  P  per 
cent  of  that  total  energy  represents  losses  which  may  be  assumed  to 

be  reducible  or  waste,  the  average  cost  of  such  losses  is  increased 

/          p  \ 
by  Q  (2  —  JQQJ  per  cent.     The  difference  in  average  unit  total 

cost  and  average  unit  waste  cost  is  then  JQQ  (  1  —  TQQ)  Ce,  or 

per  cent  of  average  cost 
1  _L  _v_  1  _,    __ 

100  100 

at  A  of  total  amount  of  energy  delivered  at  B.  '  (5) 

For  example,  if  1,000  kw.  were  transmitted  over  a  line  for  1  hr. 

with  a  loss  of  6  per  cent  —  if  10  per  cent  of  that  1,000  represents 

reducible  losses  or  energy  wasted  beyond  the  end  of  the  line 

P  =  10  Q  =  6 

If  energy  at  A  costs  .01  per  kilowatt-hour  the  average  cost  per 

kilowatt-hour  delivered  at  B  including  losses  in  AB  but  not 

fixed  charges  on  line  =    (l  +  y^j  .01  =  .0106  per  kilo  watt  hour 

(from  Eq.  3). 
The  average  cost  of  energy  waste  beyond  B  is 

1+  2-  -01  =-01114  (from  Eq.  4). 


ENERGY  COST  35 

The  difference  is  .00054  which  is  5.1  per  cent  of  the  average. 
The  same  percentage  of  increase  would  be  obtained  from  Eq.  5. 


li-—) 

V     100  / 


10    '100 
_  540 

100  +  6  106 

It  must  be  noted  that  the  above  deals  with  the  average  cost  at 
A  of  energy  delivered  at  B.  In  order  to  determine  the  average 
cost  at  B  of  energy  delivered  at  B  the  charges  on  the  line  AB  must 
be  included.  These  should  be  practically  proportional  to  W 
and  hence  would  average  the  same  per  unit  for  both  useful  load 
and  for  losses. 

The  above  merely  establishes  the  fact  of  the  increased  cost  of 
waste  energy  and  must  not  be  considered  as  indicative  of  the 
actual  amount  of  that  increase.  In  determining  this  a  number  of 
variable  quantities  must  be  considered: 

1.  The  demand  charge  for  waste  energy  at  any  point  should  be 
figured  by  the  second  method  given  under   that  heading,  i.e., 
making  up  the  demand  charge,  for  any  load,  of  charges  pro- 
portional to  the  amount  of  that  load  at  the  time  of  peak  load  on 
station,   transmission   cable,    substation,   etc.     The   reason   for 
this  is  seen  when  it  is  considered  that  the  demand  cost  on  the 
station,  for  example,  is  considered  proportional  to  the  station 
peak.     If  that  peak  can  be  reduced  by  eliminating  waste  energy 
(without  reducing  the  useful  load)   the  demand  cost  will  be 
likewise  reduced  in  proportion.     This  line  of  argument  does  not 
apply  to  useful  load  since,  for  this,  the  company  is  receiving  a 
monetary    return.     The    actual    demand    cost    should    be    dis- 
tributed to  such  loads  by  the  more  equitable  third  method.     In 
other  words,  a  reduction  of  losses  at  off-peak  time  would  not 
affect  the  actual  demand  cost  of  the  station  nor  the  distribution 
of  that  cost  among  the  useful  loads.     A  reduction  of  useful  load 
at  off  peak  would  reduce  the  revenue  and  hence  increase  the  cost 
to  other  loads. 

2.  The  example  above  uses  the  energy  charge  as  a  single  total 
charge  per  kilowatt-hour  and  the  loss  at  a  definite  percentage. 
Since  the  percentage  loss  varies  as  the  load  varies  during  the  day 
the  increase  in  cost  will  not  depend  on  the  percentage  losses  at 
maximum   load.     The   variation   in   loss   will   affect   both   the 
demand  charge  and  the  kilowatt-hour  charge  and  each  in  a 
different  proportion. 


36  ECONOMICS  OF  ELECTRICAL  DISTRIBUTION 

3.  If  several  loads  of  different  sizes  and  characteristics  are 
considered  the  problem  is  further  complicated. 

4.  If  lines  are  operated  under  the  most  economical  load,  the 
losses  may  be   considered   as  useful  load  rather  than   waste. 

As  has  been  stated  before,  the  theory  here  explained  can  also 
be  applied  to  determination  of  the  additional  cost  of  low  power 
factor,  of  load  increases  (in  some  cases)  and  other  similar  prob- 
lems. It  would  appear  that  each  unit  of  energy  comprising  any 
load  might  be  conceived  as  having  a  different  cost.  In  dealing 
with  increases  in  useful  or  saleable  load  we  have  the  further 
consideration  of  economical  loading  of  lines  so  that  it  would  not 
be  correct  to  say  that  an  increase  in  load  always  costs  more  per 
unit  than  the  average  unit  cost  before  the  increase.  In  dealing 
with  losses,  however,  a  large  percentage  of  loss  may  be  assumed 
to  cost  more  per  unit  than  a  small  percentage.  This  idea  of 
increased  cost  of  losses  and  waste  energy  is  an  important  one  to 
bear  in  mind  in  all  economical  studies,  since  usually  such  a  study 
is  fundamentally  the  establishing  of  the  most  economical  relation 
between  cost  of  energy  loss  and  other  costs. 

This  chapter  is  intended  to  be  more  a  suggestion  as  to  how  the 
problem  of  cost  of  energy  and  energy  losses  may  be  studied 
rather  than  a  complete  solution  or  a  recommended  method  to 
cover  all  cases.  It  may  be  easily  seen  that  the  study  of  energy 
cost  might  be  carried  to  an  almost  infinite  degree  of  refinement. 
Many  economical  studies  are  general  for  any  part  of  a  system  and 
are  intended  to  cover  a  period  of  time  well  into  the  future.  For 
these  cases  a  very  detailed  determination  of  energy  cost  would 
not  seem  necessary.  If  the  more  accurate  costs  are  once  deter- 
mined however,  they  can  easily  be  averaged  for  a  more  general 
problem.  Lack  of  time  and  of  the  necessary  data  will,  in  many 
cases,  limit  the  study  to  more  or  less  approximate  results.  It  is 
hoped,  however,  that  the  methods  and  principles  here  explained 
will  give  a  good  idea  of  the  nature  of  the  problem.  Detailed 
analyses  of  energy  cost,  when  data  is  available,  will  well  repay 
the  effort  expended.  Even  a  more  approximate  study  with  these 
theories  in  mind  will  indicate  the  relative  costs  of  energy  at 
various  parts  of  the  system,  of  various"  types  of  loads,  and  the 
relation  between  the  cost  of  waste  and  useful  energy. 


CHAPTER  V 
LOAD  CHARACTERISTICS 

POWER    FACTOR — BALANCE    FACTOR — DEMAND    FACTOR — DI- 
VERSITY   FACTOR — LOAD    FACTOR — EQUIVALENT    HOURS 

The  consideration  of  the  cost  of  energy,  in  the  previous  chapter 
brought  forth  some  of  the  quantities  that  are  used  in  analyzing 
the  character  of  a  load  and  its  relation  to  other  loads.  It  was 
also  indicated  that  there  are  many  kinds  of 'loads  and  that  each 
one  of  these  can  be  expressed  in  terms  of  its  characteristics.  It  is 
proposed  here  to  define  and  briefly  comment  on  some  of  these 
terms:  power  factor,  balance  factor,  load  factor,  demand  factor, 
diversity  factor,  and  on  equivalent  hours.  When  available  the 
definition  of  each  one  of  these  terms  will  be  taken  from  the 
" Standardization  Rules"  of  the  A.  I.  E.  E. 

Power  Factor. — "The  ratio  of  the  power  (cyclic  average  as 
defined  in  No.  26)  to  the  volt-amperes.  In  the  case  of  sinusoidal 
current  and  voltage,  the  power  factor  is  equal  to  the  cosine  of 
their  difference  in  phase."  (Power  in  an  alternating-current 
circuit.  The  average  value  of  the  products  of  the  coincident 
instantaneous  values  of  the  current  and  voltage  for  a  complete 
cycle,  as  indicated  by  a  wattmeter.) 

Since  the  date  of  this  issue  of  the  rules  there  has  been  con- 
siderable discussion  as  to  the  best  definition  for  power  factor. 
However  for  the  purpose  of  this  book  the  above  general  definition 
is  sufficient.  From  the  point  of  view  of  the  designer  of  an 
economical  line  the  power  factor  is  usually  determined  in  advance 
by  the  load  at  the  end  of  the  line  and  it  is  not  within  his  province 
to  change  its  value.  The  interesting  point  to  him  is  really  the 
kilovolt-amperes  his  line  has  to  carry,  as  it  is  the  current  of  a  load 
that  determines,  on  account  of  the  energy  losses,  not  only  the 
economical  loading  of  an  old  line,  but  the  economical  size  of  wire 
for  a  new  line  destined  to  carry  a  given  known  load.  It  is  evi- 
dent, however,  that  while  making  studies  in  economical  handling  of 
loads  the  effects  of  power  factor  will  be  most  forcibly  brought  to 
view  and  that  the  desirability  of  high  power  factor  and  general 
power-factor  improvement,  particularly  on  lines  carrying  large 
amounts  of  power  loads,  will  be  shown  to  be  most  imperative 
for  increased  economy. 

37 


38  ECONOMICS  OF  ELECTRICAL  DISTRIBUTION 

Balance  Factor. — While  balance  factor  has  never  been  posi- 
tively denned  it  should  be  a  means  of  expressing  the  divergence 
between  an  unbalanced  load  on  a  polyphase  circuit  and  the  same 
load  when  perfectly  balanced.  While  it  has  been  assumed 
throughout  this  book  that  loads  were  balanced,  it  is  evident  that 
many  problems  will  arise  where  it  will  be  necessary  when  solving 
for  economical  design,  to  make  allowance  for  such  unbalance. 
Large  single-phase  loads  on  polyphase  circuits  will  often  result  in 
this  necessity. 

Demand  Factor. — "The  ratio  of  the  maximum  demand  of  any 
system  or  part  of  a  system,  to  the  total  connected  load  of  the 
system,  or  of  the  part  of  system  under  consideration." 

(The  demand  of  an  Installation  or  System  is  the  load  which  is 
drawn  from  the  source  of  supply  at  the  receiving  terminals 
averaged  over  a  suitable  and  specified  interval  of  time.  Demand 
is  expressed  in  kilowatts,  kilovolt-amperes,  amperes,  or  other 
suitable  units.) 

(The  Maximum  Demand  of  an  Installation  or  System  is  the 
greatest  of  all  the  demands  which  have  occurred  during  a  given 
period.  It  is  determined  by  measurement,  according  to 
specifications,  over  a  prescribed  time  interval.) 

Demand  factor  is  therefore  the  expression  of  the  relation 
between  apparent  load  or  connected  load  and  the  largest  actual 
load  that  will  be  expected  at  any  time  on  an  installation.  For 
example,  if  a  house  is  wired  for  30  outlets  each  using  a  40-watt 

lamp  and  the  greatest  number  of  these  operating  at  one  time  is 

9 
nine,  the  demand  factor  is  ™  =  .3  or  30  per  cent.     On  the  other, 

oU 

hand  if  a  service  is  wired  for  a  range  of  5.5-kw.  capacity  and  if 
at  any  time  this  range  is  operated  with  all  the  elements  turned 
on  the  demand  factory  becomes  unity. 

Diversity  Factor. — "The  ratio  of  the  sum  of  the  maximum- 
power  demands  of  the  subdivisions  of  any  system  or  parts  of  a 
system  to  the  maximum  demand  of  the  whole  system  or  of  the 
part  of  the  system  under  consideration,  measured  at  the  point  of 
supply." 

Here  we  express  a  load  relation  between  various  loads  of  the 
same  or  of  different  demand  factors  and  other  characteristics. 
For  instance,  if  we  take  10  houses  each  having  the  30  outlets  as 
above  and  the  same  demand  factor  of  .3,  the  total  demand  will 

10  X  .3  X  30  X  40 

not  be — — or  3.6  kw.  but  some  smaller  amount 

1000 


LOAD  CHARACTERISTICS  39 

due  to  the  fact  that  the  maximum  demand  of  all  the  houses  are 
not  simultaneous.     Therefore  if  the   maximum  load  of    these 

Q  & 

houses  taken  together  is  1.8  kw.  the  diversity  factor  is  -  -  =2. 

l.o 

Also  taking  several  electric  ranges  as  above  the  diversity  factor 
may  be  .3  while  each  range  at  some  time  or  another  will  be 
operating  at  a  demand  factor  of  unity. 

Similarly  the  diversity  factor  between  transformers,  between 
substations,  between  lines  can  be  obtained. 

Load  Factor. — "The  load  factor  of  a  machine,  plant  or  system. 
The  ratio  of  the  average  power  to  the  maximum  power  during  a 
certain  period  of  time.  The  average  power  is  taken  over  a  certain 
period  of  time,  such  as  a  day,  a  month,  or  a  year,  and  the  maxi- 
mum is  taken  as  the  average  over  a  short  interval  of  the 
maximum  load  within  that  period. 

"In  each  case,  the  interval  of  maximum  load  and  the  period 
over  which  the  average  is  taken  should  be  definitely  specified, 
such  as  a  'half-hour  monthly'  load  factor.  The  proper  interval 
and  period  are  usually  dependent  upon  local  conditions  and  upon 
the  purpose  for  which  the  load  factor  is  to  be  used." 

Since  in  economical  design  annual  costs  are  generally  the  basis 
of  analysis  it  is  evident  that  the  period  of  time  to  be  used  will  be 
1  year.  In  this  book  equivalent  hours  have  been  used  for  deter- 
mining energy  losses  over  a  line  as  shown  below. 

Equivalent  Hours. — It  is  usually  convenient  in  studying  a 
given  type  of  load,  such  as  residence  lighting  for  example,  to 
obtain  the  energy  losses  per  year  in  terms  of  the  load  carried. 
By  "load  carried,"  the  peak  load  for  the  year  will  be  meant, 
since  it  is  for  that  load  that  the  size  of  the  wire  and  transformers 
must  be  determined.  Since  the  energy  loss  over  a  line  is  depen- 
dent on  the  square  of  the  current  at  any  time  it  is  evident  that 
the  total  loss  is  not  proportional  to  the  load  factor  since  the  load 
factor  is  determined  from  the  average  load  and  hence  involves 
only  the  first  power  of  the  load  at  any  time.  The  computation 
of  total  energy  loss  in  terms  of  peak  load  must  be  based  on  the 
square  of  the  loads  at  any  time.  It  is  therefore  convenient  for 
this  purpose  to  determine  for  each  of  the  various  classes  of  load 
considered  a  quantity  which  has  been  called  the  "equivalent 
hours." 

"Equivalent  hours"  may  be  defined  as  "the  average  number 
of  hours  per  day  which  it  would  be  necessary  for  the  peak  load 


40 


ECONOMICS  OF  ELECTRICAL  DISTRIBUTION 


of  the  year  to  continue  in  order  to  give  the  same  total  energy 
loss  as  that  actually  given  by  the  variable  load  throughout  the 
year."  It  is  a  quantity  which,  if  multiplied  by  the  loss  at  peak 
load  on  any  line  gives  the  average  loss  per  day  over  the  year.  If 
this  average  daily  loss  is  then  multiplied  by  365  the  total  yearly 
loss  in  kilowatt-hours  is  obtained.  This  times  the  cost  of  energy 
per  kilowatt-hour  gives  the  annual  cost  of  energy  loss.  It  is 
evident  that,  if  the  equivalent  hours  for  any  load  and  the  peak 
demand  of  the  year  are  known,  the  total  yearly  cost  for  losses 
would  be  PR  X  t  X  365  X  Ce. 

where  I  is  the  current  at  peak  load, 

R  is  the  resistance  of  the  circuit, 

t  is  the  equivalent  hours, 

Ce  is  the  cost  of  energy  per  kilowatt-hour. 

The  quantity,  equivalent  hours,  is  also  of  use  in  determining 
the  cost  of  energy  per  kilowatt-hour  previously  discussed  since 
the  cost  of  energy  losses  is  a  component  of  that  cost. 

If  the  characteristic  curve  for  any  type  of  load  were  available 
for  a  whole  year,  the  sum  of  the  squares  of  the  current  for  each 
hour  taken  from  that  curve  times  the  resistance  of  the  conductor, 
would  give  the  total  yearly  loss  in  watt  hours.  This  however 
would  be  a  tedious  computation  and  in  most  cases  impracticable. 
It  is  usually  sufficiently  accurate  to  obtain  characteristic  curves 

TABLE  1 


Circuit, 

Jan. 

Feb. 

Mar. 

Apr. 

May 

June 

July 

Aug. 

Sept. 

Oct. 

Nov. 

Dec. 

number 

1 

190 

180 

180 

175 

160 

135 

120 

125 

160 

175 

180 

195 

2 

200 

180 

175 

165 

145 

140 

110 

130 

200 

195 

205 

210 

3 

175 

165 

150 

140 

120 

125 

90 

115 

160 

165 

180 

185 

4 

185 

175 

180 

150 

140 

140 

105 

130 

175 

185 

200 

210 

5 

155 

155 

155 

150 

135 

130 

100 

130 

160 

175 

180 

195 

6 

150 

140 

135 

125 

110 

100 

90 

80 

120 

135 

150 

150 

7 

245 

225 

210 

200 

190 

180 

120 

140 

225 

245 

285 

285 

8 

235 

225 

225 

210 

200 

185 

150 

205 

250 

280 

280 

280 

9 

225 

225 

215 

225 

175 

175 

135 

170 

215 

235 

240 

250 

10 

185 

185 

185 

175 

150 

140 

130 

140 

170 

185 

225 

240 

11 

125 

120 

115 

110 

105 

105 

90 

100 

120 

125 

135 

150 

12 

130 

120 

110 

105 

100 

100 

85 

100 

125 

135 

145 

130 

Total  

2,200 

2,095 

2,035 

1,930 

1,730 

1,645 

1,325 

1,565 

2,080 

2,235 

2,  05 

2,480 

Average  of 

the  12.... 

183 

175 

170 

161 

144 

137 

111 

130 

173 

186 

200 

207 

LOAD  CHARACTERISTICS 


41 


for  each  month  or  at  least  a  typical  characteristic  curve  applica- 
ble to  any  month  with  allowance  for  the  variations  in  the  peak 
from  month  to  month.  An  example  of  an  actual  calculation  of 
equivalent  hours  on  residence  lighting  circuits  will  indicate  a 
method  which  can  be  followed  in  such  computations. 

The  characteristic  variation  of  the  load  on  a  residence-lighting 
circuit  from  month  to  month  was  obtained  from  Table  1  of 
maximum-current  readings  for  each  month  during  1  year  on  12 
typical  circuits  in  various  districts  and  with  various  loads. 


250 


ZZ5 


c 
'•150 


125 
100 


FIG.  4. — Average  of  monthly  maximum  loads  on  twelve  typical  lighting  circuits. 

The  curve  (Fig.  4)  plotted  from  the  above  average  values  indi- 
cates clearly  the  variation  of  the  monthly  peak  loads  on  a  typical 
circuit.  Since  the  peak  for  the  year  occurs  in  December,  the 
peak  for  any  month  may  be  expressed  as  a  fraction  of  this  yearly 
peak  as  follows: 


Jan. 
0.884 

Feb. 
0.845 

Mar. 
0.820 

Apr. 

0.778 

May 
0.695 

June 
0.662 

July 
0.536 

Aug. 
0.628 

Sept. 
0.835 

Oct. 
0.898 

Nov. 
0.965 

Dec. 
1.00 

Hence,  for  example,  the  maximum  current  for  February  equals 
.845  multiplied  by  the  maximum  current  for  the  year,  etc. 

For  convenience  the  PR  loss  for  any  month  may  be  considered 
equal  to  the  loss  for  a  typical  day  in  the  month  times  the  number 
of  days  in  the  month,  since  the  variation  of  load  from  one  month 
to  the  next  is  not  enough  to  warrant  more  detailed  computation. 

The  PR  loss  in  a  feeder  for  any  one  day  would  be  very  nearly 
equal  to  the  sum  of  the  squares  of  the  current  readings  for  each 


42  ECONOMICS  OF  ELECTRICAL  DISTRIBUTION 

hour  during  the  day  multiplied  by  R.  For  a  great  part  of  the 
day,  however,  the  load  on  a  lighting  circuit  is  very  light,  the  heavy 
load  and  hence  most  of  the  loss  occurring  within  a  few  hours  in 
the  evening.  By  adding  the  squares  of  the  hourly  current 
readings  throughout  a  day  and  dividing  by  the  square  of  the  maxi- 
mum current  for  the  day  a  figure  is  obtained  which  represents 
the  number  of  hours  for  which  the  peak  load  for  the  day  would 
have  to  be  carried  steadily  to  produce  the  same  I2R  loss. 

For  a  strictly  accurate  calculation,  enough  data  should  be 
available  to  determine  the  shape  of  the  load  curve  on  a  typical 
circuit  for  one  typical  day  for  each  month  in  the  year.  If  this 
is  not  possible,  however,  a  fairly  accurate  approximation  may  be 
arrived  at  if  the  load  curves  for  only  one  or  two  months  are  avail- 
able. If  the  equivalent  number  of  hours  at  peak  load  for  a 
typical  day  for  these  months  is  determined  the  equivalent  hours 
for  the  other  months  may  be  calculated  more  or  less  accurately 
by  making  them  proportional  to  the  number  of  hours  between 
sunset  and  about  10  P.  M.  allowing  a  little  additional  time  in 
the  winter  months  for  the  morning  lighting  peak.  The  figures 
obtained  in  the  present  case  for  these  monthly  equivalent  hours 
at  peak  load  will  be  found  in  the  first  column  of  the  accompany- 
ing table. 

The  loss  for  any  day  in  a  month  is  proportional  to  the  square 
of  the  maximum  current  for  the  day  multiplied  by  the  equivalent 
hours  per  day  at  peak  load  as  obtained  above.  If  now  we  assume 
that  the  average  daily  peak  will  be  95  per  cent  of  the  peak  for 
the  month,  the  loss  for  this  day  is  proportional  to  the  square  of 
the  maximum  current  for  the  month  multiplied  by  .952  multi- 
plied by  the  equivalent  hours  per  day  at  daily  peak  load  as  deter- 
mined above.  The  monthly  peak  however  is  equal  to  a  certain 
fraction  of  the  yearly  peak  as  shown  before.  Hence  the  loss 
for  this  day  in  terms  of  the  yearly  peak  is  proportional  to  the 
maximum  current  for  the  year,  squared,  multiplied  by  this 
fraction  for  the  month,  squared,  multiplied  by  .95 2,  multiplied 
by  the  equivalent  hours  per  day  at  daily  peak  load.  If  this 
figure  is  multiplied  by  the  resistance  of  the  circuit,  R,  the  actual 
loss  is  obtained.  For  example,  the  equivalent  hours  per  day  in 
terms  of  daily  peak  load  as  determined  for  February  are  4J£. 
The  peak  for  February  is  .845  of  the  yearly  peak.  Hence  for  a 
typical  day  in  February  the  P  R  loss  equals  (yearly  maximum 
current)2  X  .8452  X  .952  X  4.5  X  R. 


LOAD  CHARACTERISTICS 


43 


The  total  yearly  loss  would  be  the  sum  of  the  daily  losses 
thus  obtained.  An  average  of  the  figures  for  each  month  would 
then  give  the  average  loss  per  day.  Since  the  quantity  (yearly 
maximum  current)  2  X  R  is  the  loss  due  to  the  yearly  peak  load, 
the  average  number  of  hours  per  day  which  that  load  must  con- 
tinue may  be  obtained  by  averaging  the  other  factors  entering 
into  the  computation  of  average  daily  loss  in  terms  of  yearly 
peak.  These  are  the  quantities  (equivalent  hours  per  day  at 


daily  peak  load)  X 


X  95'    as  determined  for 


each  month.     The  resulting  average  is  the  value  of  "  equivalent 
hours  "  for  that  load. 

The  following  table  shows  the  data  worked  out  from  the  above 
example  of  residence  lighting. 

TABLE  2 


1 

« 

> 

3 

1   • 

1 

3 

<u 

?|i 

+3 

a 

CM 

1  X  3 

S    03 

h 

o 

J5*  ^ 

1 

>      03 

til 

i 

1 

fl   ^ 

LH 

1 

w^-2 

8^ 

v~-^ 

January  

5H 

.s 

«4 

.781 

4.10 

February  

4;Hz 

,i 

>45 

.714 

3.21 

March 

sK 

E 

.20 

672 

2  185 

April  

&A 

78 

.606 

1.67 

May.  ...           

2^^ 

.6 

>95 

483 

1  088 

June 

13/ 

e 

>62 

438 

767 

July.. 

2 

i 

>36 

.287 

.574 

August                 

.e 

>28 

395 

987 

September 

ol/ 

£ 

>35 

697 

2  263 

October  

4 

.£ 

>98 

.806 

3.224 

November  

5 

.c 

65 

.931 

4.655 

December  . 

clZ 

1  C 

0 

1  00 

5  500 

Total  

30  223 

Average 

2  519 

2.519  X  (.95) 2  =  2.27  eq.  hr.  per  day  at  peak  load. 

The  above  indicates  that  for  the  example  used  of  purely  light- 


44 


ECONOMICS  OF  ELECTRICAL  DISTRIBUTION 


ing  load  the  total  loss  for  the  year  will  be  the  same  as  if  the  peak 
load  were  carried  2.27  hr.  per  day  throughout  the  year. 

Corrections  for  Special  Conditions. — The  above  figures  are  sub- 
ject to  correction  under  certain  conditions.  Actual  loads  on 
circuits  were  used  and  no  allowance  was  made  for  the  normal 
increase  which  might  be  expected  on  lighting  load  due  to  addi- 
tional customers,  the  increased  use  of  current  by  old  customers, 
etc.  This  might  be  a  satisfactory  figure  for  use  in  many  cases 
but  there  might  be  occasions  in  which  a  load  showing  only 
seasonal  variations  would  be  encountered.  For  example,  if 
distribution  transformers  are  kept  loaded  nearly  to  capacity, 
new  installations  would  care  for  the  yearly  increase  and  each 
individual  transformer  would  show  nearly  the  same  load  from 
year  to  year.  In  case  the  annual  rate  of  increase  is  known,  a 
correction  can  be  applied  to  the  figures  given  for  each  month  to 
reduce  it  to  the  same  maximum  load  for  the  year.  In  the 
example  given,  the  average  yearly  increase  over  a  number  of 
years  was  found  to  be  20.3  per  cent  or  1.69  per  cent  per  month. 
Assuming  December  as  the  yearly  peak,  the  ratio  of  the  peak 
for  each  month  to  the  yearly  peak  was  corrected  by  a  proportional 
part  of  the  yearly  increase,  i.e.,  for  November  it  was  increased 
by  1.69  per  cent,  for  October  3.38  per  cent,  etc. 

TABLE  3 


Jan. 

Feb. 

Mar. 

Apr. 

May 

June 

July 

Aug. 

Sept. 

Oct. 

Nov 

Dec. 

Monthly  ratio1 

.884 

.845 

.820 

.778 

.695 

.662 

.536 

.628 

.835 

.898 

.965 

1.00 

Correction  fac- 

tor   

1.177 

1.160 

1.143 

1.126 

1.119 

1.102 

1.085 

1.068 

1.051 

1.034 

1.017 

1.00 

Corrected  ratio 

1.040 

.981 

.937 

.876 

.778 

.729 

.581 

.571 

.878 

.930 

.980 

1.00 

Monthly  Ratio 


peak  load  for  month 
peak  load  for  year 


unconnected. 


The  above  indicates  that  the  actual  corrected  peak  on  such  load 
is  in  January  but  not  enough  difference  will  be  introduced  to 
necessitate  a  revision  of  the  figures  to  that  basis. 

If  now  the  equivalent  hours  are  computed  on  the  basis  of  the 
above  ratios  a  new  figure  is  obtained  for  a  loading  with  no  yearly 
increase.  In  this  case  the  equivalent  hours  thus  corrected  are 
computed  to  be  2.65  instead  of  2.27  as  determined  for  actual 
loading,  with  a  yearly  increase. 

A  further  correction  may  be  applied  if,  as  in  case  of  transform- 
ers, the  full-load  capacity  is  to  be  used  in  studying  losses  rather 


LOAD  CHARACTERISTICS  45 

than  the  actual  load  carried.  If,  for  example,  on  the  above 
circuits  the  transformers  were  carrying,  on  an  average,  89  per 
cent  of  their  full-load  capacity,  the  equivalent  hours  based  on 
connected  capacity  would  be  2.65  X  .892  =  2.07.  That  is  to 
say,  the  year's  loss  in  energy  would  be  equal  to  the  full  load 
current  on  the  transformer  carried  2.07  hr.  per  day  throughout 
the  year. 

Other  corrections  may  be  necessary  to  meet  particular  con- 
ditions. The  above  will  give  an  indication  of  how  such 
corrections  should  be  applied. 

The  matter  of  equivalent  hours  should  be  carefully  studied  and 
as  accurate  figures  as  possible  obtained  for  various  classes  of  load. 
The  values  will  vary,  naturally,  for  different  sections  of  the 
country  as  well  as  for  different  localities  in  the  same  section  or  on 
the  same  system.  The  habits  of  a  community,  as  to  hours  of 
rising  and  retiring,  etc.,  will  affect  the  value  for  lighting  loads. 
On  power  loads,  of  course,  the  nature  of  the  industry  will  be  a 
controlling  factor.  In  general  for  the  cases  coming  within  the 
experience  of  the  writers  the  values  lie  within  the  range  given 
below : 

Power  load — from  0  to  10  eq.  hr. 
Residence  lighting — from  2  to  3  eq.  hr. 
Store  lighting  (small) — from  2  to  3  eq.  hr. 
Store  lighting  (large) — from  2  to  5  eq.  hr. 
Street  lighting — from  5  to  10  eq.  hr. 

Relation  between  Load  Factor  and  Equivalent  Hours. — It  is 

interesting  to  find  what  the  relation  between  load  factor  and 
equivalent  hours  is,  especially  as  one  or  the  other  quantity  may  be 
available  in  a  problem,  while  the  other  is  necessary  for  the 
solution  at  hand.  For  our  purpose  it  will  be  found  particularly 
useful  in  making  an  approximate  determination  of  equivalent 
hours  if  the  load  factor  is  known. 

Limits  may  be  established  within  which  the  value  of  the 
relation  between  load  factor  and  equivalent  hours  will  lie  in  all 
cases. 

The  extreme  cases  are  as  follows: 

I.  The  peak  load  is  on  for  a  short  time  only.  The  remainder 
of  the  load  curve  is  flat  for  the  rest  of  the  day.  See  Fig.  5. 

In  this  case  the  amount  of  the  continuous  load  divided  by  the 
momentary  peak  gives  the  load  factor. 


46 


ECONOMICS  OF  ELECTRICAL  DISTRIBUTION 


The  IZR  loss  =  Iav.*R  X  24  =  (LF  X  Imax)2  R  X  24. 


Equivalent  hours  at  peak  load  = 


X 


X  24 


(LF)2  X  24. 

II.  The  peak  load  is  continuous  for  a  part  of  the  day — the 
load  is  0  thereafter.  See  Fig.  6. 

In  this  case  the  load  factor  is  the  number  of  hours  the  load  is 
on,  divided  by  24. 

^MAXIMUM  LOAD 


1 

L 

5| 

-  24-  HOURS       " 
FIG.  5. 

^.MAXIMUM  LOAD 

•^1 

L 

-J 

!<•-• 

FIG.  6. 

^-MAXIMUM  LOAD 


24-  HOURS 
FIG.  7. 


The  PR  loss  =  I*maxR  X  (LF  X  24). 

Equivalent  hours  =  LF  X  24. 

III.  For  any  intermediate  arrangement  of  load.     See  Fig.  7. 

At  any  time,  tnt  let  the  value  of  /„  =  pnlmax- 


The  load  factor  = 


PR  loss  =  f(pnImax)2Rdt. 
Equivalent  hours  =  Spn*dt. 


(6) 


LOAD  CHARACTERISTICS  47 

It  is  evident  from  the  nature  of  the  curves  that  for  any  load 
factor  Cases  I  and  II  are  limiting  cases  since  no  greater  dis- 
tribution of  load  can  be  obtained  than  Case  I  and  no  greater 
concentration  than  Case  II.  The  rule  can  be  stated  as  follows: 

"For  any  given  load  factor,  the  corresponding  value  of 
equivalent  hours  will  be  somewhere  between  the  limits  of  (load 
factor  X  24)  and  (load  factor)2  X  24.  , 


CHAPTER  VI 
GENERAL  EQUATION 

KELVIN'S  LAW — GENERAL   METHOD   OF   SOLVING   PROBLEMS — 
PRESENTATION  OF  RESULTS 

The  previous  chapters  have  dealt  largely  with  the  data  neces- 
sary for  the  economical  study  of  distribution  problems,  and  the 
methods  of  obtaining  that  data.  The  cost  of  material  and  labor, 
the  annual  charges  on  these  items,  the  unit  cost  of  energy  for 
different  loads  and  the  annual  cost  of  energy  losses  have  all  been 
taken  up  in  some  detail.  Once  the  data  is  collected,  there  still 
remains  the  problem  of  so  utilizing  it  as  to  obtain  the  most 
economical  conditions  for  the  line  or  lines  under  consideration. 
Also,  means  must  be  found  for  so  exhibiting  the  results,  by 
equations,  graphs,  tables,  etc.,  that  they  will  be  convenient  of 
application  to  present  problems  and  to  future  similar  problems 
and  subject  to  easy  revision  with  changing  prices. 

Although  every  problem  of  this  nature  that  is  considered  will 
present  certain  characteristics  of  its  own  which  make  it  different 
from  all  others,  there  are  certain  underlying  principles  and 
methods  of  procedure  which  are  applicable  to  all.  A  brief  dis- 
cussion of  these  will  be  given  here.  A  good  understanding  of 
these  general  methods  will  simplify  the  study  of  their  appli- 
cation to  particular  problems,  which  will  follow  in  subsequent 
chapters. 

Kelvin's  Law. — To  Sir  William  Thompson  (Lord  Kelvin)  is 
generally  attributed  the  basic  study  of  economical  conduction  of 
electrical  currents.  In  1881  he  expressed  the  principle  that 
"the  most  economical  size  of  copper  conductor  for  the  trans- 
mission of  electrical  energy  would  be  found  by  comparing  the 
annual  interest  on  the  money  value  of  the  copper  with  the  money 
value  of  the  energy  lost  in  it  annually  in  the  heat  generated  in  it 
by  the  electric  current  .  .  .  Contrary  to  a  very  prevalent 
impression  and  belief,  the  gage  to  be  chosen  for  the  conductor  does 
not  depend  on  the  length  of  it  through  which  the  energy  is  to  be 
transmitted.  It  depends  solely  on  the  strength  of  the  current 
to  be  used  supposing  the  cost  of  the  metal  and  of  a  unit  of  energy 


GENERAL  EQUATION  49 

to  be  determined."  In  expressing  this  mathematically,  the  total 
annual  cost  was  expressed  as  the  sum  of  the  fixed  charges  on  the 
conductor  and  the  cost  of  energy  loss.  The  size  of  wire  for  which 
this  would  be  a  minimum  was  then  determined,  being  that  size 
for  which  the  two  component  charges  are  equal.  What  is 
generally  known  as  Kelvin-' s  Law  has  been  formulated  from  this, 
i.e.,  that  the  most  economical  size  of  conductor  is  that  for  which 
the  annual  charge  on  the  investment  is  equal  to  the  annual  cost 
of  energy  loss.  Under  modern  conditions  with  the  use  of  alter- 
nating currents,  wide  range  of  voltages,  large  variety  of  wire 
sizes,  with  and  without  insulation,  etc.,  an  indiscriminate  use  of 
Kelvin's  Law  as  thus  stated  is  liable  to  lead  to  considerable  error. 
It  will  apply  strictly  only  to  problems  for  which  the  cost  of 
conductor  supports  can  be  neglected  (or  is  directly  proportional 
to  the  size  of  wire),  when  the  cost  of  any  size  of  wire  is  directly 
proportional  to  the  cross-sectional  area  of  the  copper,  when  no 
transformers,  condensers,  or  other  equipment  need  be  considered 
and  when  the  cost  of  energy  loss  is  inversely  proportional  to  the 
wire  size.  Needless  to  say,  few  problems  could  be  included  under 
the  above.  In  most  cases  it  is  necessary  to  revert  to  Kelvin's 
original  method  which  was  to  determine  an  expression  for  the 
total  annual  cost  and  from  this  to  determine  the  most  economical 
condition  desired.  This  will  include  not  only  the  investigation 
of  the  most  economical  wire  size  but  also  of  the  most  economical 
voltage,  the  most  economical  route,  the  most  economical  size 
and  spacing  of  transformers,  and  other  similar  questions.  From 
this  can  likewise  be  determined  the  actual  advantage,  in  dollars, 
of  one  installation  over  another,  where  it  would  be  economical  to 
change  from  one  type  of  installation  to  another,  and  a  great 
many  other  extremely  useful  considerations. 

General  Equation. — A  general  expression  for  total  annual 
cost  which  will  be  applicable  to  most  of  the  problems  in  electrical 
distribution  lines  can  be  set  down.  Naturally  each  of  the  items 
included  will  be  somewhat  different  for  different  problems  and 
all  problems  will  not  include  all  of  these  items.  The  symbol  g 
will  be  used  throughout  to  indicate  percentage  of  fixed  charges 
on  investment.  It  will  vary,  of  course,  with  different  kinds  of 
property.  In  this  general  equation,  g  will  be  used  as  a  general 
symbol,  i.e.,  the  expression  (g  X  a  quantity)  indicates  that  the 
annual  charges  rather  than  the  first  cost  are  considered.  Then 
we  have: 


50  ECONOMICS  OF  ELECTRICAL  DISTRIBUTION 

Total  annual  cost  =  g  (cost  of  right-of-way) 

+  g  (cost  of  poles  and  fixtures  or  under- 
ground ducts  in  place) 
+  g  (cost  of  conductors  in  place) 
+  g  (cost  of  transformers  and  transformer 

equipment  installed) 

H-  g  (cost  of  any  special  equipment  used) 
+  cost  of  maintenance,  inspection,  testing, 

etc. 

+  cost  of  annual  energy  loss  on  line 
+  cost  of  annual  energy  loss  on  transformers 
and  other  equipment.  (7) 

Individual  cases  may  produce  other  charges  which  must  be 
added  but  the  above  is  characteristic. 

Units. — In  some  specific  problems,  such  as  the  comparison  of 
economy  between  two  sizes  of  conductor  for  some  particular 
location  and  definite  load,  actual  values  for  voltage,  resistance, 
costs  of  materials  and  energy  and  other  constants  could  be  intro- 
duced at  once  into  this  equation  and  the  total  annual  cost  of  each 
installation  determined  in  dollars.  In  the  usual  case,  however, 
it  is  desirable  to  make  the  study  more  general,  covering  a  more 
or  less  wide  variation  in  conditions,  so  that  it  can  be  utilized  to 
reduce  computation  on  future  problems  or  in  the  determination 
of  standards  for  a  given  class  of  installations.  For  this  purpose 
it  is  advisable  to  represent  as  many  of  the  quantities  as  possible 
by  symbols  and  to  carry  these  symbols  through  the  computations 
as  far  as  possible.  This  also  facilitates  revision  of  the  formulas, 
graphs,  etc.,  if  a  change  in  prices  makes  this  advisable.  For 
example,  the  following  are  some  of  the  symbols  most  commonly 
used  in  this  book: 

TABLE  4 

W  =  load  in  watts  P  =  resistivity    of    conductor 

kw.  =  load  in  kilowatts  material 

E  =  voltage  cos  6  =  power  factor 

I  =  amperes  A  =  cross-sectional  area  of  con- 

R  =  resistance  of  circuit  ductor. 

X  =  inductance  of  circuit  /l/o''  TC  =  weight     of     conductor     per 

r  =  unit  resistance  of  conduc-  unit  length 

tor  T  =  transformer  size  in  kilovolt- 

x  =  unit    inductance    of  con-                     amperes 
ductor 


GENERAL  EQUATION  51 

TABLE  4  (Continued) 

Rt  =  equivalent  resistance  of  trans-  g  =  per  cent  fixed  charges,  (interest 

formers  taxes,  depreciation,  etc.) 

Cr  =  cost  of  right-of-way  per  unit  I  =  equivalent  hours 

Ccu  =  cost  of  copper  per  pound  V  =  per  cent  voltage  drop 

Csr  =  cost  of  stringing  conductors  P  =  per  cent  power  loss 
Ce  =  cost  of  energy  (subscript  1,  2, 
"3,  etc.,  indicate  variations) 

In  any  given  case,  most  of  these  quantities  will  be  fixed  by  the 
conditions  of  the  problem  and  may  be  considered  as  constants 
so  that  the  general  equation  for  annual  cost  may  usually  be 
reduced,  in  its  final  form,  to  one  containing  only  two  or  three 
variables,  such  as  load  in  kilowatts  or  amperes,  cross-sectional 
area  of  wire,  power  factor,  or  equivalent  hours.  Examples  of 
this  will  be  shown  later. 

Presentation  of  Results. — The  equation  for  total  annual  cost 
once  obtained,  there  still  remains  the  question  of  getting  from  it 
the  information  desired  in  the  best  possible  form  for  convenient 
use.  It  is  found  that  there  are,  in  general,  three  ways  in  which 
it  is  convenient  to  accomplish  this.  Conditions  of  the  problem 
and  the  results  desired  will  determine  which  one  of  these  is  most 
applicable  in  any  case. 

1.  The  actual  annual  cost  in  dollars  may  be  plotted  in  a 
curve  or  series  of  curves.  Where  the  equation  for  annual  cost 
is  expressed  in  more  than  one  variable,  such  as  size  of  wire  and 
load,  for  example,  one  of  these  variables  must  be  held  constant 
in  plotting  any  one  curve.  Sufficient  number  of  such  curves 
must  then  be  plotted  to  show  the  required  variation  in  that 
quantity.  In  this  way  a  series  of  curves  may  be  obtained,  for 
example,  one  curve  for  each  standard  size  of  wire,  showing  the 
relation  between  annual  cost  and  load  on  a  line  with  that  size. 
An  example  of  this  method  is  shown  in  Fig.  8.  When  the  expres- 
sion for  annual  cost  contains  more  than  two  such  variables,  how- 
ever, this  method  is  usually  not  applicable, 
v  2.  In  some  cases  with  three  variables  entering  into  the  total 
annual  cost  the  following  method  will  be  founpl  convenient. 
Suppose,  for  example,  the  annual  cost  depends  on  a  variable 
value  for  equivalent  hours,  variable  load  and  variable  wire  size. 
If  the  expression  for  annual  cost  for  two  standard  sizes  of  wire 
are  equated,  the  resulting  equation  plotted  between  load  and 
equivalent  hours  shows  the  dividing  line  between  economy  for 


52 


ECONOMICS  OF  ELECTRICAL  DISTRIBUTION 


one  size  or  the  other.     The  accompanying  figure  (Fig.  9)  shows 
such  a  series  of  curves  as  determined  for  three-phase  secondary 


15  ZQ  25  30 

Load  Density  in  Kw.  per  1000  feet 

FIG.  8. — Annual  cost  per  1,000  ft.  of  3  No.  4  secondaries.     (Includes  fixed  charges 
on  wire  in  place  and  energy  loss.) 


0       10      W      30      40       50      60      70       80      90      100      HO     120      130      WO      150     160     170 
K_   Hours  per  Week 

FIG.  9. — Most  economical  size  of  wire  for  three-phase  secondaries  (small  power 

'    loads). 

for  a  certain  type  of  load.  Any  point  lying  between  two  curves 
indicates  economy  for  the  corresponding  size  of  wire.  This 
method  has  the  disadvantage  of  not  exhibiting  quantitative 


GENERAL  EQUATION 


53 


economy.  The  results  are  qualitative  only,  since  the  nearer 
the  point  lies  to  either  limit  of  the  area  the  less  the  relative  dif- 
ference between  the  cost  with  the  size  of  wire  indicated  and  the 
next  adjacent  size. 

3.  A  third  method  gives  results  that  are  neither  quantitative 
nor  qualitative.  Also,  its  application  is  usually  somewhat 
limited.  In  certain  cases,  however,  it  is  preferable  to  any  other 
method.  Where  the  expression  for  annual  cost  contains  several 
variables,  if  these  can  all  be  reduced  to  terms  of  two  variables 


£ 

c 

Ib 

Q 


Load  Density  in  Kw.per  1000 feet 
FIG.   10. — Most  economical  voltage  drop  on  lighting  secondaries. 

(as  percentage  voltage  drop  and  load,  for  example)  and  the  first  v 
derivative  of  the  resulting  expression  with  respect  to  one  of  these 
variables  be  set  equal  to  0,  the  equation  thus  obtained  will  give 
the  most  economical  relation  between  the  .two  variables  (as 
most  economical  voltage  drop  for  any  load  under  the  given  con- 
dition). The  accompanying  figure  gives  an  example  of  a  curve 
derived  in  this  manner  (Fig.  10). 

As  was  suggested  before,  it  will  be  found  of  considerable 
advantage  to  prepare  the  .equations  from  which  the  final  curves 
are  plotted  in  as  general  terms  as  possible,  i.e.,  with  as  many  of  the 
constant  quantities  as  possible  represented  by  symbols.  It  is, 
then,  a  comparatively  easy  matter  to  revise  the  curves  to  meet 
changing  prices  of  material  and  labor,  or  other  conditions  of 
load,  voltage,  power  factor  or  materials  of  construction  than 
those  originally  contemplated. 


54  ECONOMICS  OF  ELECTRICAL  DISTRIBUTION 

In  some  cases  it  may  be  found  useful  to  develop  the  results 
in  numerical  tables  or  sometimes  merely  by  a  simple  formula. 
Such  cases  are  not  the  most  usual,  however,  and  the  above 
methods  will  probably  be  found  sufficient  for  most  purposes. 
On  some  more  extensive  problems,  all  three  methods  of  exhibiting 
data  will  be  used. 

In  future  chapters,  a  number  of  problems  met  with  in  practice 
will  be  taken  up  and  their  treatment  in  accordance  with  these 
methods  will  be  described.  Some  of  the  details  necessarily 
omitted  in  the  previous  discussion  of  these  methods  will  be 
brought  out  in  the  individual  problems  and  the  procedure  hereto- 
fore described  in  general  terms  will  be  actually  carried  through. 


CHAPTER  VII 
POWER  LOSS  AND  VOLTAGE  DROP 

CHARTS  FOR  SIMPLIFIED  SOLUTION  FOR  POWER  Loss  AND  VOLTAGE 

DROP 

Power  Loss. — In  most  of  the  solutions  of  problems  in  economy 
appearing  in  this  book,  the  power  loss  is  introduced  as  a  function 
of  the  load,  the  wire  size,  the  equivalent  hours,  etc.  The  reason 
for  this  is  evident  from  the  nature  of  the  methods  used  and  the 
results  desired.  There  are  often  cases,  however,  where  the 
power  loss  alone  is  wanted.  There  will  be  given  in  this  chapter 
simple  curves,  with  their  derivation,  which  enable  the  power 
loss  to  be  quickly  solved  in  the  great  majority  of  problems. 
The  most  usual  case  of  power  loss  is  that  due  to  the  resistance 
of  the  conductor,  i.e.,  the  PR  loss.  In  high-tension  trans- 
mission lines,  leakage  losses  and  corona  losses  become  important. 
Charging  current  also  has  an  effect  on  PR  loss.  Such  problems, 
however,  are  comparatively  rare  in  the  work  of  most  engineers 
and  warrant  special  treatment  when  encountered.  Methods  of 
solving  for  corona  loss,  leakage,  etc.  are  given  in  the  handbooks 
and  elsewhere  and  a  discussion  of  them  is  beyond  the  province  of 
this  work.  In  the  greater  majority  of  problems,  the  PR  loss  is 
all  that  need  be  considered. 
If  D  =  length  of  line  in  feet, 
W  =  the  load  at  the  receiver  end  in  watts  ( =  kilowatts  X 

1,000), 

E  =  the  receiver  voltage, 

A  =  cross-sectional   area   of   conductor   in   circular   mils, 
p  =  resistivity  of  conductor  material  in  ohms  per  mil  foot, 
cos  8  =  the  power  factor  of  the  load  W. 

PR  loss  =  (•=- — — )    ^-  X  2  watts  (for  single-phase)  (8) 

\Hi  COS  u/      A. 

I        W        \ 2  oD 
PR\oss  =  (-7=^ -)    "—  X  3  watts  (for  three-phase)         (9) 

\\/3Jl/  COS  "/     A. 

If  P  =  per  cent  power  loss  in  terms  of  power  delivered, 
JP  _       2W2PD          _1 
100       (E  cos  BY  A  X  W 

P  = 

(E  cos 

\  55 


56  ECONOMICS  OF  ELECTRICAL  DISTRIBUTION 

_P  TPpP  1 

100       (E  cos  0)2A  A  F 


For  the  same  load,  same  voltage  between  conductors,  and  the 
same  conductor  size  the  loss  with  single-phase  is  twice  that  with 
three-phase. 

If  it  is  desired  to  consider  the  load,  voltage  and  power  factor 
at  the  source  instead  of  at  the  receiver,  if 

Wf  =  load  at  source  in  watts, 
E'  =  voltage  at  source, 
cos  6'  =  power  factor  at  source. 

The  line  loss  in  percentage  of  load  at  source 
D/  20QW'pD    ,,       .     ,      , 

P     =(f°rsmgle-phase) 


or  three-phase)  (13) 

For   copper   conductor  p  =  10.8   approximately.     If  load  is 
expressed  in  kilowatts  (kw),  the  formula  becomes — 

2.16Xl06Xfcw.Z>,,       •     ,      , 
P  =  ~     -TCT-T-^TI — (for  single-phase)  (14) 


1.08  X  106  Xkw.D 


These  formulas  are  comparatively  easy  to  use.  However  it  is 
believed  the  work  may  be  somewhat  simplified,  especially  where 
a  large  number  of  such  computations  are  to  be  made,  by  use  of 
the  accompanying  chart  (Fig.  11).  The  use  of  this  chart  reduces 
the  computation  to  a  simple  multiplication  of  round  numbers. 

The  chart  is  plotted  as  follows: 

A  series  of  circular  arcs  (with  the  center  at  0,  0)  are  drawn, 
each  representing  a  given  voltage.  Voltages  range  from  0  to 
150,  but  as  will  be  shown  below,  the  same  arcs  may  be  used  for 
any  voltage.  Diagonal  straight  lines  are  drawn  through  0, 
0  at  various  slopes,  each  representing  a  given  power  factor.  It 
is  evident  that  the  abscissa  of  the  intersection  of  any  arc  with 
any  diagonal  gives  the  corresponding  value  of  E  cos  0. 

For  each  standard  wire  size  considered  a  curve  is  now  plotted 

j) 
between  r  —  n/i  QQQ  anc^  E  cos  #  giymg  f°r  anv  voltage,  power 

factor,  and  wire  size  the  percentage  power  loss  per  kilowatt  per 


POWER  LOSS  AND  VOLTAGE  DROP 


57 


1,000  ft.  Curves  for  other  sizes  of  wire  can  be  easily  added  if 
desired.  The  upper  part  of  these  curves  was  drawn  to  a  con- 
densed scale  to  give  a  greater  range  of  values  in  the  more  un- 
usual cases.  The  curves  here  shown  are  for  three-phase.  The 


1 


FOR  USE  IN  DETERMINING  POWER  LOSS  IN  PERCENT 
OF  DELIVERED  POWER  ON  THREE  PHASE  LINES 
For  single  phase  multiply  scale  readings  by  2 

To  Use :  I.  -Reduce  voltage  to  an  equivalent  secondary 
—  voltage  within  Hie  limits  of  the  chart  — 
(OtolSOvolts)  by  dividing  by  a  convenient 
ratio,  2, 10,  200,  etc. 
2. -Locate  intersection  of  voltage  arc  with 

power  factor  diagonal  (E cos  O) 
3.- Intersection  of  this  ordinate  with  curve 
forgiven  wire  size  gives  percent  power 
loss  on  scale  at  left. 

4.-D/vide  scale  read  ing  by  the  square  of  ratio 
used  in  1,  and  multiply  by  distance  in 
-thousands  of  feet  and  loat 


RESUL  T=  Total  percent  power 
o\o  /oss  on  line 


Ib  50  75  100 

EcosQ 

FIG.  11. — Power  loss  curves. 


\15 


power  loss  for  single-phase  would  be  twice  the  values  shown  for 
three-phase. 

The  use  of  these  curves  is  as  follows: 

1.  Reduce  the  voltage  considered  to  any  equivalent  secondary  voltage 
within  the  scale  of  the  chart  (less  than  150  volts)  by  dividing  by  some 
convenient  transformation  ratio  such  as  2,  4,  10,  20,  200,  400,  1,000,  etc. 


58  ECONOMICS  OF  ELECTRICAL  DISTRIBUTION 

2.  Locate  intersection  of  voltage  arc  and  power  factor  diagonal. 

3.  The  intersection  of  the  vertical  through  this  point  with  the  curve  for 
the  proper  wire  size  gives  the  value  of  P  on  the  scale  at  left. 

4.  Divide  this  scale  reading  by  the  square  of  the  transformation  ratio 
used,  to  give  percentage  power  loss  per  kilowatt  per  1,000  ft.  for  the  given 
voltage.     Multiply  by  the  number  of  kilowatts  and  by  the  length  of  line 
in  thousands  of  feet  for  total  percentage  power  loss  in  terms  of  power 
delivered.     (If  line  is  single-  phase  multiply  this  quantity  by  2.) 

Take  for  example  the  following  problem  : 

Load  —  1,200  kw.  three-phase 

Power  factor  —  75  per  cent 

Voltage  at  receiver  —  1,400 

Wire  size  —  3  No.  0 

Distance—  8,000  ft. 

4  400 
'4Q     =110  (transformation  ratio  =  40) 

.38  3 

Power  loss  =  ^  x  ^  X  WW  X  8  =9.12  per  cent  or  109.4  kw. 

Voltage  Drop.  —  In  studying  the  economy  of  a  line,  it  must  not 
be  forgotten  that  the  element  of  good  service  is  also  important. 
Good  service  depends  largely  on  good  regulation  which,  in  turn 
depends  a  great  deal  on  the  voltage  drop  of  the  line.  Often, 
a  line  economically  loaded  will  have  too  great  a  voltage  drop. 
If  it  is  artificially  regulated,  the  cost  of  the  regulator  enters  into 
the  consideration  of  economy.  In  some  cases  the  use  of  large 
conductors  without  a  regulator  may  be  more  economical  than 
smaller  conductors  with  a  regulator. 

The  computation  of  voltage  drop  is  apt  to  be  a  tedious  opera- 
tion if  carried  out  often.  Several  means  of  simplifying  this  have 
been  published,  such  as  the  well-known  Mershon  Diagram  and 
the  Dwight  chart. 

When  relating  to  high-tension  lines,  the  problem  of  voltage 
drop  involves  consideration  of  line  capacity,  etc.  These  prob- 
lems are  usually  of  sufficient  importance  to  warrant  detailed 
computation.  Formulas  for  this  are  given  in  the  handbooks 
and  elsewhere. 

El  =  E2  cosh  \/ZY  +  h  \/Z7Y  sinh  \/ZY 


Ii  =  72  cosh  \ZT  +  (E2/Z/Y)  sinh 
Where  EI  and  E2  are  voltages  from  phase  to  neutral  at  the  send- 
ing and  receiving  ends  respectively;  Z  is  the  impedance  per  wire; 
Y  is  the  admittance  from  phase  wire  to  neutral. 
For  medium  and  low-voltage  problems,  it  is  usually  sufficient 


POWER  LOSS  AND  VOLTAGE  DROP  69 

to  consider  inductive  reactance  and  resistance  only.  The  method 
and  charts  for  computing  voltage  drop,  given  below,  are  based 
on  these  quantities.  They  are  derived  from  the  equation  ordi- 
narily used  for  such  computations,  i.e. 

JL     -  "V(ff  cos  0  +  VZRID)2  +  (E  sin  0  +  A/3  X  ID)2  -  E 
100  E 

(for  three-phase  lines) 


,  \2  ,    /  .         .   V3  X  IDY     t 

cos  0  +     —  ^  -  I  +  (sin  0  +       —  ^  --  )  -1 

Where  V  =  per  cent  voltage  drop  in  terms  of  receiver  voltage, 
R  =  resistance  in  ohms  per  foot, 
X  =  inductive  reactance  in  ohms  per  foot, 
D   =  distance  from  source  to  receiver  in  feet, 
I     =  current  per  wire. 

But 

100  W  D       100  -v/3  RID 
P  =  per  cent  power  loss  =  77^  --  T^VT  =  -  FJ  — 

(E  cose)2  A          E  cos  e 

(see  above).    (17) 
Substituting 


icfo  =  \/( 


cos  9  +  fo?cos  e     +  sin 


=  cos 


7  =  B  =  10°  cos  e   / /  i  4.     P\  2+  /tan  g  +     P  X\* -  10° 

(19) 

(The  above  computation  is  for  three-phase  but  the  resulting 
expression  for  B  is  the  same  for  single-phase.) 

B  then  is  a  quantity  expressing  the  relation  between  per  cent 
voltage  drop  and  per  cent  power  loss.  It  depends  on  the  power 

/X\ 
factor  (cos  6),  size  and  spacing  of  conductors  (-p)  and  percent 

\  Kit 

power  loss  (P),  but  is  independent  of  load  or  voltage.  It  is 
evident  that  if  B  is  known  the  per  cent  voltage  drop  may  be 
easily  obtained  from  the  per  cent  power  loss  by  the  simple  relation 

V  =  BP  (20) 

B  may  be  plotted  as  shown  in  the  accompanying  curves  (Fig. 
12)  (a)  and  (b).  Since  the  variation  of  B  with  P  is  not  great 
excepting  for  high-power  factor  and  large  conductors,  three 


60 


ECONOMICS  OF  ELECTRICAL  DISTRIBUTION 


3.0 


'g'.v/i 

Power  Factor^eO,  70,80,90% 
Percent  volts  drop -of delivered  volts,  1.5 


power 


,"  r 

"   TO  USE '.-Determine  value  of  x/r  for  si: 
andspacingofwire  used-from 

curves  on  right. 1 

Determine  value  of  "B '  for  this 
value  of  x/r  from  proper  curve 
on  left.        |  | 

Multiply  bu  per  cent  power 

lo&s  (from  other  curves)  for Q  t, 

percent  volts  drop  (V)   V-PB 


2.5 


)       20       W       GO      80 
Spacing  in  Inches 


FIG.  12a—  Values  of 


"B 

Values  of  "B"=y/P  for 

=30,  95,  100  % 

V=Percent  volts  drop  -of  delivered  volts 
P=     »      power  hss  -of     »     power  \. 


TO  USE '.-Determine  value  ofX/rforsne 
'<o        and  spacing  of  wire  used-  from  curves 
on  right 

Determine  value  of  "B  "for  this  value 
of  x/r  from  proper  curve  on  left: 


Mutt  i  ply  by  percent  power  loss(from 
er  curve  st  for 


of  her  curve  st  for  percent  volts  drop 


1.5 
"B" 
FIG.  126.— Values  of  "B  " 


20       40       60      80 
Spacing  in  Inches 


POWER  LOSS  AND  VOLTAGE  DROP 


61 


fixed  values  of  P  were  chosen  which  cover  the  usual  range  of 
problems,  i.e.,  1  per  cent,  10  per  cent  and  20  per  cent.  For 
each  power  factor,  a  curve  between  B  and  X/R  is  plotted  for 
each  of  these  values  of  P.  Other  values  of  P  may  be  interpolated 


Percent  of  delivered  power  tost" 
jper  kilowatt  per  1000  feet  o  ver 
3  phase  line    ~ 
For  single  phase  line  multiply 
figure  given  by  curve  by  2 

To  oMain  percent  voltage  drop,  use 
these  curves  in  conjunction  with 
quantity  "B'g/ven  by  figure  tea 


10  80 

Power  Factor 

FIG.   13a. — Power  loss  curves. 


if  necessary.  The  value  of  X/R  is  shown  directly  by  the  curves 
on  the  right  for  standard  sizes  of  conductor  and  any  spacing. 
Use  of  Curves.  —  The  use  of  the  curves  is  as  follows:  Locate 
the  point  on  the  X/R  curve  for  the  given  wire  size  and  spacing. 
(For  three-phase  unequal  spacing  use  the  equivalent  spacing,  S  = 
Pass  across  the  sheet  to  the  left  on  the  horizontal 


through  this  point  until  it  intersects  the  curve  for  the  given  power 


62 


ECONOMICS  OF  ELECTRICAL  DISTRIBUTION 


factor  and  value  of  P.  The  corresponding  value  of  B  is  found 
on  the  scale  at  the  bottom.  If  this  is  multiplied  by  the  value  of 
P  computed  from  Fig.  11,  the  desired  value  of  V  (per  cent  voltage 
drop)  is  obtained. 

Example. — In  the  example  given  above  for  the  determination 
of  Pj  let  the  spacing  between  the  wires  be  28  in.,  28  in.,  56  in. 


L§. 

105 


\ 


PERCENT  OF  DELIVERED  POWER  LOST  PER  KILOWATT 
PER  IOOO  FEET  OVER  3  PHASE'  LINE 

For  single  phase  line  multiply 
figure  given  by  curve  by  2 


Use  curve  marked  "44OO  Volts  "for  44,  OOO  440O 
44OJ  2^000,  23OO,  220 and  110  Volfs. 
Use  curve  marked  "4600  Volte  "for  4G,  OOO.  4GOO. 
460, 23,000, 2300,  230 and  1/5 Volts. 

Use  curve  marked  "SOOOVolts  "forSqOOOjSOOO, ' 
SCO,  25000,  2SOO,  2SO  and  I2S  Volts,  with 
proper&cale.  \ 

To  obtain  percent  voltage  drop,  use 

these  curves  in  conjunction  with 

quantity  "B"given  by  other  curves . 


4.8 


0.48   I.J 


10  80 

Power  Factor 

FIG.  136. — Power  loss  curves. 


64 

105 


T^i  0.56  2.24 


The  equivalent  spacing  =    V28  X  28  X  56  =  35  in. 
Interpolating  between  the  70  per  cent  and  80  per  cent  power- 
factor  curves 

B  =  1.18 

V  =  1.18  X  9.12  =  10.75  per  cent  or  473  volts. 


POWER  LOSS  AND  VOLTAGE  DROP 


63 


The  curves  given  here  are  general  and  may  be  used  for  any 
voltage,  load,  length  of  line  and  spacing  between  conductors. 
The  range  of  wire  sizes  and  power  factor  is  more  limited  but 
the  curves  could  be  easily  extended  to  cover  any  desired  value. 
For  everyday  use  on  the  problems  arising  on  any  given  system  less 
general  curves  may  be  derived  from  these  which  still  further 
simplify  computations.  As  an  example,  Fig.  13  (a)  and  (6) 
shows  a  set  of  curves  plotted  for  work  on  4,600-volt  lines.  Three 
curves  are  shown  for  each  wire  size  giving  a  range  from  4,400  to 


ine 


per  1 


^ 


~*y 


0.6 


-4600W1--80%  Power  Factor -Spacmg27"a 


mf 


400 


dOO 


1200 


1600          2000  2400 

Load  in  Kilowatts 


2800 


3200        3600 


FIG.  14. — Curves  showing  per  cent  voltage  drop  (of  delivered  voltage)  per  1,000 

ft.  of  line. 

5,000  volts.  For  any  power  factor  (lower  scale)  the  scale  on  the 
left  gives  the  per  cent  power  loss  per  kilowatt  per  1,000  ft. 
Figure  13  (6)  shows  how  scales  for  other  voltages  can  be  added. 

Figure  14  shows  another  special  curve  giving  per  cent  voltage 
drop  per  1,000  ft.  of  line  for  4,600  volts,  three-phase,  80  per  cent 
power  factor,  27-in.  spacing. 

Figure  15,  called  "Load  Curve  for  Power  Lines",  gives  the 
distance  to  which  any  load  can  be  carried  on  a  three-phase, 
4,600-volt  line  at  80  per  cent  power  factor  with  a  10  per  cent  drop 
in  voltage. 

Figure  16  is  a  similar  curve  for  single-phase  lines  with  a  power 
factor  of  95  per  cent. 


64 


ECONOMICS  OF  ELECTRICAL  DISTRIBUTION 


Figure  17  is  a  load  curve  for  220-volt,  three-phase  secondary 
with  10  per  cent  drop. 

Figure  18  is  a  load  curve  for  22^i2-volt,  single-phase  second- 
ary with  3  per  cent  drop. 


36,000 


32000 


28000 


»      Load  transmitted  by  3 phase 
OL     3wirej46OO  volt- lines  with" 


^    IO%  voltage  drop 
^^  Power  factor 80%  -28  Spacing 
^  v  Load  concentrated  at  end 


NO  TE^The  values  KW  for  any  , 
dkfter  voltage  drop  are  ven. 
nearly  proportional  to 

os^fbr  10  %  drop      

Foroth^r  voltages,  KWis 


proportioned  to  the  square, 
of  the  voltae* 


Safe  Current  Carry  ing 


24000 


20000 


16000 


12000 


8000 


4000 


800  1200 

Load  Trcmsmittedj 

FIG.  15. — Load  curves  for  power  lines. 

It  is  a  comparatively  simple  matter  to  derive  any  such  special 
curves  desired  by  use  of  the  general  curves  for  power  loss  and  B 
given  above. 

Approximate  Method  for  Secondaries. — For  low-voltage 
problems,  such  as  for  secondaries,  an  approximate  determination 


POWER  LOSS  AND  VOLTAGE  DROP 


65 


is  usually  as  accurate  as  one  more  detailed.     It  will  be  found 
that  the  expression 

-j  =  volts  drop  per  ampere  =  R  cos  6  +  X  sin  8       (21) 

while  an  approximation,  is  sufficiently  accurate  for  most  problems 
of  this  class. 

36,000r 


32,000 


28,000 


24,000 


20,000 


9000 


8000 


15,000 


re.ooo 


8000 


4000 


100  200  .    300  400  500  600 

Load  in  Kilowatts 

FIG.  16. — Load  curves  for  single-phase  lines. 


As  an  example  for  the  use  of  this  expression,  the  curves  in  Fig. 
19  (a.b.c)  are  given.  The  load  considered  is  residence  lighting 
on  three- wire,  11?^20~volt  secondary.  The  average  per  residence 
is  assumed  to  range  from  135  to  200  watts.  Two  different  lengths 


66 


ECONOMICS  OF  ELECTRICAL  DISTRIBUTION 


1200 


Horsepower-  Load  Concentrated  at  End  of  Secondary 
0      20      40     60      80      100     120     140-160     180     200    220    240    260    280    300    320    340    560 


i         i         i         i         i  i 

Load  Transmitted  by  3  Phase,  3  Wire,  22O  Volf  Secondary 
with  10  %  Vol+a&e  Drop.          \ 
Power  Factor  80  %  -28  "spacing 


Where  W=  Load  in  waffs 

A  =  Cross-sectional  area  of  conductor 
V  =  Percentage  voltage  drop         \         \ 
D  -  Distance  from  source  to  receiver  in  feef 
B.and  C-  Cons  tan  ts 
I 


For  other  va/ues  of  voltage  drop  W/s  approximately 
proportional  fo  V  (B  varies  somewhat  with  V)  — f 
For  other  vo/f age  e  W  varies  with  £  z 


0      40      80     120      160    200    240    280     320    360    400    440    480     520    560    600    640    680  720 
Horsepower-  Load  Uniformly  pfstributed  Transformer  at  End 

FIQ.  17. — Load  curves  for  power  lines. 


6000 


Load  transmitted  by  3  w'ire,  .224/nz  volt  secondaries 


. 
with  3%  voltage  drop     II 

ower  Factor  95  %-Loaduni  form/t  d 


D=  Length  of  line  one  way  fo. 
concentrated  load,  transformer 
atend 

length  of  line  one-wau  L    — 
wQr 


distributed  had  transformer; 
at  end 


..length  of  line  one  way 


distributed  had,  transformer 
erf center 


10       12       14       16       18      20      22      24      26      2&      30      32      34     36 
Kilowatts  Transmitted 


FIG.  18. — Load  curves  for  secondaries. 


POWER  LOSS  AND  VOLTAGE  DROP 


67 


of  average  span  are  taken  as  shown.  The  total  voltage  drop 
from  the  transformer  to  the  end  of  the  secondary  is  computed 
by  multiplying  the  number  of  customers  at  each  pole  by  the 
spans  between  that  pole  and  the  transformer  and  using  the  sum 
of  these  multiplications  to  find  the  drop  in  volts  with  the  proper 


^(No.Services  x  No.Sections) 
10  20 

V 

IT 
16 


L 


TO  USE: 

_  I.- Multiply  the  number  of  _ 
services  from  any  pole  by  the 

number  of  sections  ft 

rransforme 


actions  trorn 
products} 


2- Add  all  these  _ 
3.  -  Curve  for  appropriate  length 
~*~of  section  and  average  load  ~ 
per  service  gives  total  volts 
drop  to  end  of  secondary. 


NOTE:- For  3 #e  (load  balanced) use  bo   ' 
and  right  hand  scales 


For  2*6  use  bottom  and/efthandscafes. 
or  top  and  right  hand  scales. 


10      20      30      40      50      60      10      80      90      100     110    120 
^  (No.  Serv'icesx  No.Sections) 


FIG.  19a.  —  Voltage  drop  curves  for  110-220  volt  secondaries  (No.  6  wire). 


curve  and  scale.  The  scales  are  so  arranged  that  for  three- wire 
secondary,  i.e.,  220  volts  with  load  balanced  the  bottom  and 
right-hand  scales  should  be  used.  For  two-wire  secondary,  110 
volts,  the  bottom  and  left-hand  scales  or  the  top  and  right-hand 
scales  give  the  required  result. 

For  example  consider  a  secondary  as  shown  below: 


68 


ECONOMICS  OF  ELECTRICAL  DISTRIBUTION 


Sections  150  ft.,  average  load  135  watts 

First  pole  1X1=1 

Second  pole  2X2=4 
Third  pole 

Fourth  pole  1X4=4 
Fifth  pole  3  X  5  =  15 
Sixth  pole  1X6=6 


^(No.Services  x  No.Sections) 

20  36  40  50 


Obtained  from  The 
approximate  formula  :- 
Volts  drop  per  ampere 
=RcosG+XsinQ 


LL 


UL 


LL 


7 


Z 


services  from  any  pole 
-by  the  number  of  sections 
from  transformer 

2- Acfdallthese  products   

3-  Curve  tor  appropriate 

length  of  section  and  average 
—  load  per  service  gives  ratal - 
volt£c-  '  •  - 
•seconc/dru 


nd^t 

ed)cise 

nd  scales. 


NO  TE:  -  For  3  #4  (load  balanced)  use 
bottom  and  righ  t  hands  cales. 
For  2  #4  use  bottom  and  le  ft 
.  hand  scales  or  top  and  right . 
hand  scales 


50 


100    .  150  200 

^.  ( No .  Services  x  No. Sections) 


FIG.  196.—  Voltage  drop  curves  for  110-220  volt  secondaries  (No.  4  wire), 


Seventh  pole 
Eighth  pole  2  X    8  =  16 
Ninth  pole  3  X    9  =  27 
Tenth  pole 

Eleventh  pole  2  X  11  =  22 
Total  95 


POWER  LOSS  AND  VOLTAGE  DROP 


69 


Voltage  drop  from  curve  3.93  volts.  Which  is  higher  than  it 
should  be,  3^  volts  being  about  the  limit  to  be  used.  In  case 
the  drop  to  the  eighth  pole  is  desired. 


^ (No. Services  *  No  Sections) 
20  30  40  50 


£ 

VOLTAGE  DROP  CURVES 
16 1 Obtained  from  -Hie 


approximate  formula:- 
Volfs  drop  per  ampere 
~  Rcos  O+X&ih  B 


Multiply  thenutnbei 
^  ofservices  from  any 
pole  by  the  number  of 
sections  from  trans  - 
former  — — [ 


2.  Add  all  these  products 
3.  Curve  for  appropriate  length 
of  sect  ion  and  aver  age  load  per 
service  gives  total  volts  drop  to 
end  of  secondary 


A 'OTE:- For  3#2(load  balanced)  use  bo  from 
and  right  hand  scales. 

For  2#Z  use  bottom  and  left  hand 
scales  or  top  and  right  hand  scales  - 


50  100  150  200 

^  (No.  Services  x  No.  Sections) 

FIG.  19c.  —  Voltage  drop  curves  for  110-220  volt  secondaries  (No.  2  wire). 
TRANSFORMER 


SMER  X  3  2 

-"---!-"—---0—^—  ---°—- 


FIG.  20. 

Sum   of  first  seven  above  =  30 
Eighth  pole,  7X8  =  56 

86 
Voltage  drop,  3.56  volts. 


PART  II 


CHAPTER  VIII 
TRANSMISSION-LINE  PROBLEMS 

METHOD  OF  DETERMINING  MOST  ECONOMICAL  DESIGN  FOR  MAIN 

OR  " BACKBONE"  TRANSMISSION  LINES 

"BACKBONE"  TRANSMISSION  LINES 

In  considering  the  economical  design  of  transmission  lines  two 
general  types  of  lines  are  encountered.  These  may  be  called  for 
convenience : 

1.  " Backbone"  transmission  lines. 

2.  Secondary  transmission  lines. 

To  the  first  classification  belongs  that  line  or  system  of  lines 
which  forms  the  ''backbone"  of  any  system  large  or  small.  It 
is  usually  a  line  which  transmits  a  comparatively  large  load 
(relative  to  the  total  load  on  the  system),  to  some  considerable 
distance,  at  a  relatively  high  voltage.  Secondary  transmission 
lines  on  the  other  hand  partake  more  of  the  nature  of  distribution 
lines.  They  distribute  the  energy  from  the  central  feeding 
points,  served  by  the  " backbone"  line,  to  auxiliary  stations  from 
which  it  may  be  distributed  by  the  ordinary  distribution  lines. 
For  example,  in  a  large  system  there  may  be  several  generating 
stations  tied  together  by  a  110,000- volt  line,  "the  backbone." 
At  various  points  along  this  line,  substations  are  located,  stepping 
the  voltage  down  to  22,000  volts,  the  secondary  transmission. 
These  run  to  the  various  communities  to  be  served  where  the 
voltage  is  transformed,  in  local  substations,  to  ordinary 
distribution  voltage. 

Each  of  these  types  of  lines  present  certain  distinguishing 
characteristics  which  affect  the  method  used  in  determining  the 
most  economical  design.  The  problem  of  a  backbone  line  is 
usually  a  specific  one,  i.e.,  a  definite  load  of  known  characteristics 
is  to  be  transmitted  from  one  given  point  to  another.  Future 
small  extensions  are  not  anticipated,  these  being  cared  for  by 
the  secondary  transmission  lines.  On  the  other  hand  certain 

71 


72  ECONOMICS  OF  ELECTRICAL  DISTRIBUTION 

conditions  such  as  voltage,  wire  size,  and  span  are  not  limited 
except  by  considerations  of  economy  and  good  operating 
conditions. 

Secondary  transmission  lines  on  the  other  hand  must  care  for 
a  variety  of  loads  and  distances.  They  are  subject  to  short 
extensions  to  care  for  additional  loads.  They  are  apt  to  be  tied 
together  in  a  network.  All  these  points  must  be  considered  and 
the  best  voltage,  span,  etc.  to  fit  the  average  conditions  must  be 
adopted  as  a  standard.  Wire  sizes  may  also  be  standardized  to 
one  or  two  sizes.  Further  problems  are  then  limited  to  consider- 
ations of  the  economical  load  for  any  line,  number  of  lines  for 
any  load,  most  economical  routes,  etc.  The  problems  of  under- 
ground transmission  lines  fall  mostly  under  this  classification  but 
will  be  touched  on  separately  with  other  underground  problems. 

Naturally  the  above  division  is  somewhat  elastic.  In  some 
systems  there  are  no  true  backbone  lines,  the  high-voltage 
lines  being  so  extensive  as  to  be  similar  to  secondary  transmission 
lines.  On  the  other  hand,  in  small  systems  there  may  be  no 
secondary  transmission  lines,  one  or  two  lines  comprising  the 
whole  transmission  system.  In  such  cases  the  backbone  lines 
are  of  such  a  voltage,  load,  etc.  that  on  a  large  system  they  would 
probably  be  classed  as  secondary  transmission  or  even  distribu- 
tion. They  are  backbone  lines,  however,  in  relation  to  the  small 
system,  and  their  voltage,  wire  size,  etc.  may  usually  be  deter- 
mined by  economy  without  any  local  limiting  conditions.  In 
this  chapter,  the  problem  of  the  backbone  line  will  be  taken  up 
and  the  various  elements  affecting  its  solution  will  be  discussed. 
A  line  of  relatively  high  voltage  and  heavy  load  will  be  assumed 
but  the  same  principles  might  be  applied  to  any  backbone  line. 

Having  given  the  load  to  be  carried,  with  its  probable  future 
increase,  the  points  from  and  to  which  the  load  is  to  be  trans- 
mitted, and  the  characteristics  of  that  load,  i.e.,  its  variations 
during  the  day  and  seasons,  and  the  cost  per  kilowatt-hour  at 
the  generating  station,  the  problem  remains  to  determine  the 
most  economical  route,  the  most  economical  voltage,  the  most 
economical  wire  size  and  the  most  economical  span  and  arrange- 
ment of  supporting  structures. 

It  may  be  stated  at  once  that  no  method  of  solving  for  any  of 
these  unknown  quantities  has  yet  been  presented  which  is  simple 
and  at  the  same  time  accurate  enough  for  the  basis  of  a  final 
design.  Several  writers  have  published  approximate  methods 


TRANSMISSION-LINE  PROBLEMS  73 

of  making  such  determinations.  In  these,  however,  the  variable 
quantities  are  so  numerous  that  it  is  necessary  to  make  certain 
assumptions  for  simplicity  and  thus  the  results  obtained  are 
subject  to  considerable  question.  It  is  believed  that  these  are 
valuable  in  establishing  the  limits  of  a  problem  but  that  the 
actual  design,  especially  for  a  line  of  any  considerable  importance, 
should  be  checked  up  by  a  summation  of  actual  cost  figures  as 
compared  with  the  cost  of  several  other  possible  alternatives. 

Some  of  the  approximate  methods  are  given  in  the  references 
below.1 

In  the  paper  on  " Problems  of  220  kv.  Power  Transmission"  by 
A.  E.  Silver  in  the  A.  I.E.  E.  Proceedings,  June,  1919,  is  given  one 
of  the  most  careful  and  complete  studies  of  a  high-tension  trans- 
mission line,  from  an  economic  point  of  view,  yet  published.  In 
that  paper,  the  voltage  is  assumed,  the  economical  wire  size  is 
determined  from  consideration  of  fixed  charges  and  losses,  and 
the  most  economical  span  determined  by  a  complete  comparison 
of  cost  figures  on  actual  tower  designs. 

General  Equation. — In  this  problem,  as  is  the  case  with  most 
economic  problems,  the  design  sought  for  is,  in  general,  that  one 
for  which  the  total  annual  cost  will  be  a  minimum.  The  total 
annual  cost  may  be  expressed  as  follows,  letting  "g  X  (any 
quantity)"  indicate  that  the  annual  cost  is  to  be  used. 

Total  annual  cost  =  g  (cost  of  right-of-way) 

+  g    (cost    of   towers    and   foundations   in 

place) 

+  g  (cost  of  insulators  in  place) 
-f-  g  (cost  of  conductor  and  ground  wire  in 

place) 

+  g  (cost  of  transformers  in  place) 
+  g  (cost  of  lightning  arresters  in  place) 
+  g  (cost  of  switches  in  place) 

1  "Economic  Voltage  of  Long  Transmission  Lines,"  by  HENRY  H.  PLUMB, 
Journal  A.  I.  E.  E.,  April,  1920. 

"Transmission  Line  Design,"  by  F.  K.  KIRSTEN,  A.  I.  E.  E.  Proceedings, 
Vol.  37,  1917,  p.  685. 

"Electric  Power  Transmission,"  by  A.  E.  STILL,  p.  64,  approximate 

economical  voltage  =  A/distance  +  -  -  an  empirical  formula. 

"Notes  on  the  Calculation  of  Transmission  Lines  for  Maximum  Econ- 
omy," by  E.  BATICLE,  Revue  Generate  de  L'Electricite,  Oct.  30,  1920. 


74  ECONOMICS  OF  ELECTRICAL  DISTRIBUTION 

+  g  (cost  of  special  apparatus,  regulators, 

condensers,  etc.  in  place) 
+  g  (cost  of  substation  structures) 
+  annual  cost  of  energy  loss  on  the  line 
+  annual  cost  of  energy  loss  on  the  trans- 
formers 

+  annual  cost  of  patroling,  testing  and  other 
maintenance.  (22) 

Elements  Affecting  Costs. — Each  of  these  subdivisions  of  cost 
is  affected  by  one  or  more  of  the  variable  quantities  whose 
solution  is  being  sought,  some  in  an  extremely  complicated 
manner.  The  manner  in  which  the  costs  are  affected  will  be 
discussed  briefly. 

(a)  Cost  of  Right-of-way. — Right-of-way  may  have  a  definite 
cost  for  the  whole  line  regardless  of  tower  size  or  spacing,  in 
which  case  it  is  affected  only  by  the  route  chosen.  Again,  it 
may  vary  with  the  number  of  towers  only,  regardless  of  their 
size.  In  other  places,  it  may  vary  both  with  the  number  and 
base  dimensions  of  the  towers.  The  locality  through  which  the 
line  will  pass  and  the  value  of  land  will  determine  which  of  the 
above  is  applicable  to  the  problem  in  hand. 

(6)  Cost  of  Towers. — The  cost  of  towers  is  affected  by  a  great 
number  of  conditions  and  its  relation  to  the  economy  of  a  line 
is  a  difficult  problem.  The  type  of  tower  to  be  used,  wide  base 
or  narrow,  one,  two  or  more  circuits,  etc.,  is  of  primary  impor- 
tance. The  type  of  base,  in  turn,  depends  somewhat  on  the  right 
of-way  available.  The  number  of  circuits  depends  on  the 
economical  size  of  conductor,  the  limits  for  corona  voltage  on 
conductor  and  the  expected  increase  in  load.  The  height  of  the 
tower  is  also  a  determining  factor.  This  depends  on  the  minimum 
clearance  requirements  for  the  span,  and  the  sag  in  the  conductor 
(which  in  turn  depends  on  the  span,  wire  size  and  the  assumptions 
for  heaviest  loading).  It  also  is  affected  by  the  vertical  spacing 
between  conductors,  which  is  a  function  of  the  voltage.  The 
horizontal  spacing  (which  varies  with  the  voltage)  likewise  affects 
the  cost,  since  the  weight  and  torsional  stresses  are  increased  by 
longer  arms.  The  wire  size  influences  the  cost  of  the  tower 
since  in  most  cases  the  tower  design  depends,  to  a  considerable 
extent,  on  the  maximum  stress  allowable  on  the  conductor,  which 
is  proportional  to  its  cross-sectional  area.  The  span  is  also  a 


TRANSMISSION-LINE  PROBLEMS  75 

factor,  since  the  longer  the  span,  the  greater  the  lateral  wind 
pressure  on  the  conductors,  and  this  is  an  important  factor  in 
the  design  of  the  straight  line  towers.  It  appears  to  be  practi- 
cally impossible  to  develop  any  simple  formula  which  would  indi- 
cate the  variation  in  tower  cost  with  all  these  variable  elements. 

(c)  Cost  of  Insulators. — The  cost  of  the  insulators  increases 
with  the  voltage  but  not  in  direct  ratio  on  account  of  the  de- 
creased efficiency  per  unit  in  long  strings  of  insulators. 

(d)  Cost  of  Conductor. — The  cost  of  conductor  in  place  will  be 
very  nearly  proportional  to  its   cross-sectional  area.     It  will, 
of  course,  depend  also  on  the  material  used.     It  may  vary  some- 
what with  the  number  of  towers.     The  cost  of  ground  wire  will 
in  most  cases  be  practically  a  constant  for  the  line,  depending 
only  on  its  length,   and  number,  size  and  material  of  ground 
wires  decided  upon. 

(e)  Cost  of  Transformers. — The  transformer  cost  will  increase 
as  the  voltage  increases  (but  not  in  direct  ratio)  and  also  depends 
on  the  size  of  unit  to  be  used  and  the  type  (air-cooled  or  water- 
cooled;  secondary- voltage;  etc.).     In  most  problems  of  this  kind 
the  secondary-voltage  requirements  will  be  established  and  the 
type  and  size  of  transformer  will  be  indicated  by  the  load  and  type 
of  substation  to  be  used.     The  cost  will  then  vary  with  the 
voltage  only. 

(/)  Cost  of  Lightning  Arresters. — If  a  given  type  of  arrester  is 
chosen,  the  cost  will  vary  with  the  voltage. 

(g)  Cost  of  Switches. — If  the  type  and  size  of  switch  is  deter- 
mined by  the  load,  the  cost  will  also  vary  with  the  voltage. 

(h)  Cost  of  Special  Apparatus. — If  special  apparatus  is  needed 
to  maintain  proper  voltage  regulation,  its  cost  must  be  included 
in  determining  economy,  since  it  might  be  eliminated  if  large 
enough  conductor  or  enough  lines  were  used.  The  cost  will 
vary  with  the  size  required,  which  depends  upon  the  amount  of 
regulation,  and  also  with  the  voltage. 

(i)  Cost  of  Substation  Structures. — For  a  given  size  trans- 
former, the  cost  of  the  terminal  station  pertaining  to  the  trans- 
mission line  will  be  practically  constant,  varying  somewhat  with 
the  voltage  on  account  of  the  increased  spacing  of  conductors. 
If  regulators  or  condensers  are  used  the  cost  of  the  substation 
space  occupied  by  them  must  be  included. 

(j)  Cost  of  Energy  Loss. — The  energy  lost  on  a  transmission 
line  consists  of  PR  loss  due  to  the  resistance  of  the  conductor, 


76  ECONOMICS  OF  ELECTRICAL  DISTRIBUTION 

leakage,  and  corona  loss.  If  the  insulators  are  well  designed 
and  the  conductor  is  larger  than  the  corona  limit  for  the  voltage 
used,  the  latter  two  will  be  comparatively  small  and  in  most 
cases  may  be  neglected  in  determining  economy.  The  I2R  loss 
naturally  depends  on  the  cross-sectional  area  of  conductor,  on 
the  material  used,  and  on  the  current  transmitted.  The  current 
is  a  function  of  the  voltage,  if  the  load,  power  factor,  equivalent 
hours,  etc.  are  fixed  (neglecting  charging  current).  If  charging 
current  must  be  considered,  the  average  between  current  at 
source  and  at  receiver  will  be  sufficiently  close  approximation  in 
most  cases.  The  cost  of  the  PR  loss  can  be  computed  from  the 
PR  loss  at  peak  load,  equivalent  hours,  and  cost  per  unit  of 
energy  loss.  The  latter  two  are  usually  fixed  by  local  condi- 
tions and  are  not  variables  for  the  problem. 

The  energy  losses  on  transformers  consist  of  core  losses  and 
copper  losses.  Since  transformer  efficiencies  do  not  vary  greatly 
with  size  or  voltage,  the  variations  in  the  losses  will  not  be  con- 
siderable. In  general  these  losses  decrease  somewhat  as  the  size 
of  transformer  increases  and  increase  slightly  as  the  voltage 
increases.  The  exact  amount  of  variation  must  be  deter- 
mined for  the  particular  transformers  to  be  used.  Other  energy 
losses  will  be  encountered  if  special  apparatus  such  as  regulators 
or  condensers  are  used  and  the  cost  of  this  loss  must  be  included. 

(k)  Cost  of  Patroling,  etc. — The  cost  of  testing  insulators  will 
depend  somewhat  on  the  number  of  units  used  in  a  string  and 
hence  on  the  voltage.  The  cost  of  patroling  will  vary  with 
the  length  of  the  line  and  the  nature  of  the  country  to  be 
covered.  Other  maintenance  and  repair  costs  can  only  be  esti- 
mated and  will  probably  be  roughly  proportional  to  the  length 
of  the  line. 

Solution  by  Equation  Impracticable. — The  above  is  an  indica- 
tion of  the  various  number  of  elements  which  affect  the  cost  of 
each  of  the  factors  that  go  to  make  up  the  total  annual  cost  on 
a  transmission  line.  It  is  evident,  that,  with  so  many  variable 
quantities,  there  is  no  simple  solution  for  minimum  annual  cost, 
which  will  be  general  for  all  cases.  It  is  true  that,  for  any  given 
problem,  when  all  the  conditions  are  fixed  excepting  the  four 
chief  variables  mentioned  above,  i.e.,  voltage,  wire  size,  tower 
spacing,  and  route,  equations  can  be  written  for  each  of  the  ele- 
ments of  cost  in  terms  of  these  variables,  which  will  approximate 
quite  closely  that  cost  under  any  condition.  For  example,  the 


TRANSMISSION-LINE  PROBLEMS  77 

cost  of  a  given  type  of  transformer  can  be  represented  with  suffi- 
cient accuracy  by  the  expression 

Cost  =  Ki  +  K2  En 

where  KI,  K2  and  n  are  constants  and  E  is  the  voltage.  Some  of 
the  costs  are  very  difficult  to  represent,  the  cost  of  towers  for 
example.  If  all  these  expressions  are  determined,  however, 
and  are  combined  into  the  general  equation  of  annual  cost, 
it  will  be  found  that  this  equation  is  so  complicated  and  contains 
the  variables  in  so  many  different  powers,  that  a  solution  is 
impossible  except  by  trial.  There  is  then  no  apparent  advantage 
over  the  method  of  assuming  a  number  of  alternative  designs, 
using  definite  values  for  each  of  the  variables,  and  computing 
the  actual  annual  cost  of  each  of  these  designs.  A  comparison 
of  these  costs  will  indicate  the  most  economical. 

Method  of  Solution  Outlined. — In  order  to  systematize  the 
computation  and  to  facilitate  the  determination  of  the  effect 
on  the  total  cost,  of  changes  or  additions  to  any  design,  the 
following  method  is  suggested. 

1.  Fixed  Quantities. — As  many  as  possible  of  the    elements 
affecting  the  problem  should  be  fixed.     These  will  be: 

(a)  The  maximum  load.  The  expected  increase  should  also  be  deter- 
mined. In  case  of  more  than  one  feeding  point  each  load  must  be  considered 
separately. 

(6)  The  load  factor  and  equivalent  hours  of  the  load. 

(c)  The  power  factor  of  load. 

(d)  The  cost  of  energy  per  kilowatt  hour. 

(e)  The  length  and  cost  of  right-of-way  (at  least  approximately)  for  the 
various  possible  routes. 

(/)  The  type  of  tower  to  be  used  for  each  route. 

(g)  The  per  cent  annual  charges  applicable  to  the  various  kinds  of  property 
entering  into  a  transmission  line. 

2.  Limits   of  Problem. — It   is   essential  to  discover  approxi- 
mately the  limits  of  the  problem  so  that  time  will  not  be  wasted 
in  considering  voltages  and  wire  sizes  which  are  far  from  the  final 
result.     In  case  the  designer's  experience  is  not  sufficient  to  tell 
him  this,  the  approximate  formulas  found  in  the  references  given 
above  will  be  found  useful. 

3.  Cost    Data. — The    necessary    cost    data,    quotations,    etc., 
must  be  collected.     These  will  be: 

(a)  Cost  of  transformers,  of  the  size  determined  by  the  load  for  various 
voltages  within  the  range  of  the  problem  (more  than  one  size  may  be  neces- 


78  ECONOMICS  OF  ELECTRICAL  DISTRIBUTION 

sary  especially  if  there  are  several  feeding  points  with  different  sized  loads.) 
Core  loss  and  copper  loss  should  also  be  ascertained. 
(6)  Cost  of  switches,  of  proper  size,  for  various  voltages. 

(c)  Cost  of  lightning  arresters,  for  various  voltages. 

(d)  Cost  of  insulators,  for  various  voltages.     The  limit  in  mechanical  load 
for  any  string  of  insulators  should  also  be  determined.     In  cases  where  pin 
insulators  are   considered  in   comparison  to  suspension  type,  the  cost  of 
both  must  be  obtained.     The  cost  of  placing  insulators  should  be  included. 

(e)  The  cost  per  pound  of  conductors  of  different  materials — copper, 
copper-clad  steel,  and  aluminum  steel.     In  case  the  price  per  pound  varies 
considerably  with  the  size,  the  cost  for  a  variety  of  sizes  within  the  range  of 
the  problem  should  be  obtained.     The  cost  of  stringing  the  conductor  should 
be  estimated  or  determined  from  previous  experience. 

(/)  The  cost  of  various  sizes  of  tower  for  each  type.  If  possible  sufficient 
costs  on  towers  should  be  obtained  that  the  variation  of  cost  with  span,  wire 
size  and  voltage  may  be  determined.  This  could  be  accomplished  if  three 
voltages,  three  wire  sizes,  and  three  different  spans,  covering  the  probable 
range  of  the  problem,  are  considered.  By  plotting  curves,  the  intermediate 
values  could  be  determined  with  sufficient  accuracy.  The  cost  of  foundation 
should  be  included  with  each  tower.  Anchor  towers  and  semi-anchor  towers 
should  also  be  considered  and  similar  costs  obtained. 

(g)  The  cost  of  terminal  substations  of  the  required  size.  The  variation 
in  this  cost  with  voltage  should  be  determined,  if  possible.  In  case  other 
special  terminal  apparatus,  such  as  condensers  or  regulators  are  considered, 
the  additional  substation  cost  for  these  must  be  estimated. 

(h)  Quotations  on  special  apparatus  considered,  regulators,  condensers, 
etc.,  should  be  obtained  with  variations  in  this  cost  with  size  and  voltage. 

(i)  Annual  cost  of  maintenance,  testing,  patroling,  etc.  must  be  estimated. 

4.  Arrangement  of  Cost  Data. — After  the  data  is  collected,  it 
should  be  arranged  for  convenient  use.  This  probably  can  best 
be  done  by  means  of  curves.  Such  curves  are  illustrated  in  the 
accompanying  figures.  They  should  give: 

(a)  Annual  cost  of  transformers  (in  place)  in  terms  of  voltage  (Fig.  21). 
(6)  Annual  cost  of  switches  (in  place)  in  terms  of  voltage  (Fig.  22). 

(c)  Annual  cost  of  lightning  arresters  (in  place)  in  terms  of  voltage  (Fig. 
22). 

(d)  Annual  cost  of  insulators  per  string  in  terms  of  voltage. 

(e)  Annual  cost  per  mile  of  conductors  of  different  materials  in  place  in 
terms  of  cross-sectional  area. 

(/)  Annual  cost  of  towers  in  place.  These  can  be  arranged  (for  example) 
as  a  series  of  curves  for  each  standard  voltage  showing  for  each  standard  wire 
size  (and  material)  the  variation  of  cost  with  span. 

(g)  Annual  cost  on  terminal  substation  in  terms  of  voltage. 

(h)  Annual  cost  of  transformer  energy  losses  in  terms  of  voltage. 

(i)  Annual  cost  of  line  energy  losses  per  mile  for  each  material  of  conductor 
considered,  in  terms  of  cross-sectional  area. 


TRANSMISSION-LINE  PROBLEMS 


79 


5.  Most  Economical  Design. — It  now  remains  to  apply  these 
costs  to  determine  the  most  economical  design.  The  route  is 
usually  the  most  limited  of  all  the  variables,  there  ordinarily 
being  no  more  than  two  or  three  possible  routes  at  most.  Like- 


24,000 


65     10     15      80      85      90       95      100    105     110     115      120    125     I2>0      135     140     145     150     155 

Voltages  in  K.Y. 

FIG.  21. — Cost  of  single-phase  60-cycle  transformers. 


16,000 


12,000 


65,000       15,  00          85,000 


Switches        Cost- 

105,000         115,000        <Z5,000         Ii5,000        145,000 
Voltage 


5,000 


FIG.  22. — Cost  of  lightning  arresters  and  switches  showing  variation  of  cost  with 

voltage. 


wise,  on  any  route,  the  layout  is  somewhat  limited.  Locations 
of  anchor  towers  and  semi-anchor  towers  are  usually  more  or 
less  fixed.  On  straight  runs  the  span  can  often  be  varied.  If 
two  or  three  possible  layouts  are  made  for  each  route,  varying 
the  span  where  possible,  a  sufficient  comparison  will  be  obtained. 


80  ECONOMICS  OF  ELECTRICAL  DISTRIBUTION 

If,  now,  for  each  of  these  layouts,  a  sheet  of  curves  is  prepared, 
showing  for  each  standard  conductor  considered,  a  curve,  giving 
the  variation  in  total  annual  cost  with  the  voltage,  a  complete 
and  accurate  study  of  the  economy  may  be  made.  Not  only  will 
the  most  economical  voltage,  wire  size,  route  and  layout  be  ob- 
tained, but  the  comparative  effect  of  variation  in  these  quantities 
may  be  studied.  Curves  may  be  added  to  show  comparison  of 
two  or  more  circuits  with  a  single  circuit.  If  a  design,  chosen 
tentatively,  proves,  on  further  investigation,  to  be  impracticable 
on  account  of  corona  limit,  too  great  regulation,  etc.,  the  cost  of 
correcting  the  fault  by  a  change  in  design,  addition  of  regulator, 
etc.,  as  compared  with  one  of  the  other  designs  not  having  that 
fault,  may  be  easily  determined. 

Space  does  not  permit  dwelling  on  the  subject  of  " backbone" 
transmission  lines  more  in  detail.  Any  problem  of  this  kind  is 
one  which  will  bear  an  almost  infinite  amount  of  study.  Natu- 
rally the  detail  to  which  such  a  study  should  be  carried  will 
depend  somewhat  on  the  size  and  nature  of  the  project.  As  a 
rule,  however,  time  spent  on  an  economical  determination  is  well 
repaid.  In  many  cases  some  of  the  variables  are  limited — voltage 
may  be  determined  by  other  than  economical  considerations, 
route  and  location  for  towers  may  be  fixed,  etc.  This  simplifies 
the  problem  but  in  any  case  some  such  study  as  that  outlined 
above  is  essential  to  an  accurate  determination.  Variations  in 
the  method  will,  of  course,  be  found  to  accommodate  special 
conditions.  In  any  case,  it  is  to  be  emphasized  that  the  problem 
is  a  complicated  one  and  usually  does  not  bear  solution  by  any 
easy,  approximate  method. 


CHAPTER  IX 

TRANSMISSION-LINE  PROBLEMS 
SECONDARY  TRANSMISSION  LINES 

DETERMINATION   OF  MOST   ECONOMICAL  STANDARDS   OF   CON- 
STRUCTION, CONDUCTOR  SIZE,  LOADING,  ROUTE,  ETC.  ON 
LESSER    OR    SECONDARY    TRANSMISSION    LINES 

In  the  preceding  chapter  there  were  discussed  the  points  of 
distinction  between  " backbone"  transmission  lines  and  " second- 
ary" transmission  lines.  It  is  purposed  here  to  consider  the 
problems  arising  in  connection  with  " secondary"  transmission 
lines  and  to  indicate  their  economic  solution. 

The  " secondary"  transmission  lines  of  any  system  must  be 
considered  as  a  class  instead  of  as  one  or  more  specific  problems. 
Their  service  is  varied.  They  are  called  upon  to  feed  loads  of 
various  sizes,  and  various  power  factors  and  load  factors.  They 
may  have  a  considerable  diversity  in  lengths,  types  of  route, 
etc.  It  is  obviously  impracticable  to  consider  each  line  as  a 
special  problem.  It  is  necessary  to  adopt  certain  standards 
which  will  best  fit  all  cases  on  an  average,  allowing  some  variation 
if  necessary  in  wire  sizes,  etc.  There  then  remains  the  problem 
of  how  best  to  adapt  these  standards  to  fit  any  given  condition. 

In  this  study,  therefore,  there  are  two  separate  divisions.  The 
first  is  of  a  somewhat  similar  nature  to  that  of  the  " backbone" 
transmission  line,  excepting  that  the  problem  is  general  rather 
than  specific.  It  deals  with  the  establishment  of  standards 
which  will  best  fit  average  conditions,  such  as  a  standard  voltage, 
one  or  more  standard  wire  sizes  (the  variety  depending  somewhat 
on  the  range  of  loads  handled)  and  a  standard  span.  The 
second  division  includes  problems  relating  to  the  proper  use  and 
combination  of  these  standards.  Such  problems  as  the  most 
economical  route,  the  economical  division  of  load  among  several 
lines,  the  load  at  which  wire  size  should  be  changed  or  an  addi- 
tional line  run,  and  reconstruction  problems  are  in  this  class. 

It  would  be  impossible  in  a  reasonable  amount  of  space  to 
attempt  to  cover  in  any  detail  all  the  problems  which  might  arise 
in  connection  with  secondary  transmission  lines.  The  field  of 
study  is  very  large  and  new  problems  are  constantly  appearing. 
Some  of  the  problems  commonly  met  with  under  each  of  the  two 

6  81 


82  ECONOMICS  OF  ELECTRICAL  DISTRIBUTION 

divisions  will  be  discussed,  however,  and  the  elements  affecting 
their  solution  will  be  indicated. 

The  general  method  of  attacking  any  problem  of  this  kind  is  to 
determine  an  expression  for  annual  cost  and  then  to  discover 
under  what  conditions  that  cost  will  be  a  minimum.  Since  the 
results  are  to  be  of  general  application,  it  is  necessary  that  the 
data  used  be  general,  for  the  whole  system.  Average  costs  of 
pole  line  per  mile  should  be  used,  for  example,  rather  than  an 
estimated  cost  for  a  certain  line  over  a  given  route.  Similarly, 
the  results  obtained  should  be  displayed  in  formulas,  curves  or 
tables  which  may  be  of  general  application. 

Data  Necessary. — The  data  necessary  for  solving  such  prob- 
lems on  secondary  transmission  lines  include  the  following: 

(a)  Range  of  sizes  of  loads  considered. 

(6)  Characteristics  of  various  types  of  load  carried,  load  factor,  power 
factor,  equivalent  hours,  etc.  If  the  characteristics  of  the  general  types  of 
load  are  thoroughly  understood  any  individual  load  can  be  studied  as  a 
combination  of  several  general  types  in  different  amounts.  A  load  may 
consist  of  part  lighting,  part  power,  part  street  railway,  etc.  and  its  individual 
characteristics  are  a  combination  of  the  characteristics  of  all  these  types. 

(c)  Average  material  and  labor  costs  on  standard  construction;  average 
right-of-way  costs  if  such  figures  are  obtainable. 

(d)  Cost  of  energy  for  various  types  of  load  in  various  parts  of  the  system. 

Standard  Voltage. — The  solution  of  the  first  class  of  problems 
mentioned  above  is  very  often  limited  by  other  considerations 
than  strict  economy.  Extensive  systems  rarely  spring  into  being 
suddenly.  Usually  they  are  developments  from  comparatively 
small  beginnings.  Hence,  what  was  yesterday  the  backbone 
transmission  line  of  the  small  system,  becomes  today  part  of  the 
secondary  transmission  network  of  the  large  system.  It  happens, 
therefore,  that  the  secondary  transmission  voltage  is  rarely 
chosen  as  such,  but  is  rather  a  development  from  past  practice. 
There  is  always  the  question  of  the  economy  of  changing  the 
standard  voltage,  usually  to  one  higher.  This  can  only  be 
determined  by  a  careful  study  of  present  and  probable  future 
conditions,  taking  into  consideration  the  additional  cost  for 
transformers  and  station  equipment  and  the  cost  of  making  the 
change-over,  as  compared  with  the  saving  in  line  losses,  the  smaller 
conductor  used,  and  the  increase  in  capacity  in  the  system  with 
the  proposed  higher  voltage .  Such  a  change  can  sometimes  be  made 
economically,  where  single-phase  transformers  are  used  to  a  large  ex- 


TRANSMISSION-LINE  PROBLEMS  83 

tent,  by  changing  from  a  delta  to  a  star  connected  system.  In  cases 
where  the  voltage  can  be  chosen  in  advance  on  the  basis  of 
economy,  an  analysis  similar  to  that  suggested  for  " backbone" 
transmission  lines  will  be  necessary. 

Standard  Span. — The  standard  span  to  be  chosen  will  depend 
largely  on  the  type  of  construction  used.  Conditions  may  vary 
from  the  use  of  a  steel-tower  line  on  private  right-of-way,  to 
that  of  wood-pole  lines  along  the  highway,  carrying  distribution 
in  addition.  For  the  first,  probably  the  most  advantageous  span 
can  be  chosen  somewhat  as  outlined  for  " backbone"  lines  and 
need  not  necessarily  be  uniform  nor  the  same  for  all  lines.  For 
the  latter,  the  span  will  be  determined  partly  by  the  strength 
of  the  poles  and  partly  by  the  needs  of  the  distribution  lines. 
These  spans  can  be  fairly  uniform,  not  exceeding  a  definite  maxi- 
mum of  150  to  250  ft.  ordinarily. 

Conductor  Size. — The  question  of  standardization  of  conductor 
size  will  depend  largely  on  the  kind  of  loads  carried  and  their 
distribution  over  the  system.  In  some  cases  it  will  be  found  ad- 
vantageous to  standardize  on  one  or  two  sizes  and  care  for 
additional  load  by  more  lines.  In  others,  it  may  be  better  to  use 
a  wider  variety  of  conductors,  accommodating  the  size  to  the 
load  carried.  A  method  of  studying  conductor  economy  is  given 
below.  By  its  means,  the  most  economical  wire  size  for  any  load 
may  be  chosen  and,  under  proper  conditions,  standard  sizes  may 
be  fixed.  This  method  of  study  also  aids  in  the  solution  of 
several  of  the  problems  of  the  second  class  as  will  be  pointed  out 
later. 

Example  of  Study  of  Conductor  Economy. — Let  a  standardized 
type  of  construction  with  given  voltage  and  standard  span  be 
assumed.  The  total  annual  cost  of  such  a  transmission  line  is 
composed  of: 

1.  Annual  cost  on  construction  exclusive  of  conductor. 

2.  Annual  cost  of  conductor  in  place. 

3.  Annual  cost  of  energy  loss. 

For  this  example,  a  wood-pole  line  will  be  assumed  with  a 
maximum  span  of  175  ft.  to  accommodate  distribution. 

The  annual  cost  of  poles  and  fixtures  per  1,000  ft.  of  line  will, 
of  course,  vary  with  the  wire  size,  if  the  line  is  properly  designed 
for  safety,  probably  after  some  such  formulas  as  KI  +  K2A, 
where  KI  and  K2  are  constants  and  A  is  the  cross-sectional  area 
of  the  conductor. 


84  ECONOMICS  OF  ELECTRICAL  DISTRIBUTION 

The  annual  cost  of  the  conductor  per  1,000  ft.  will  vary  approxi- 
mately with  A  as  also  will  the  cost  of  stringing,  i.e.,  cost  in  place  = 
K3  +  K*A. 

The  annual  cost  of  energy  loss  per  1,000  ft.  will  vary  with  the 
square  of  the  load,  the  first  power  of  the  equivalent  hours,  and 
the  cost  of  energy,  and  inversely  with  the  cross-sectional  area. 

Then  the  total  annual  cost  per  1,000  ft.  of  line 


Y  =  K,  +K3  +  (K2  +  K4)A  +  Kb  (23) 

Where  t  =  equivalent  hours,  Ce  cost  of  energy  loss  per  kilowatt- 
hour,  kw  =  load  in  kilowatts 
Simplified 


Y  =  Ka  +  Kb  A  +  Kc  (24) 

Where  Ka  =  K,  +-K>  and  Kb  =  K2  +  K*,  and  Kc  =  Kb 

Most  Economical  Conductor  Size.  —  The  most  economical 
conductor  size  will  be  that  for  which  the  value  of  Y  is  a  minimum. 
This  can  be  obtained  by  setting  the  first  derivative  of  Y  with 
respect  to  A  equal  to  0 

dY  kwHC. 

dA    =  Kb       Ac      A2 


A  =  kw  J^  t  Ce  (25) 

Most  Economical  Load. — If  it  is  desired  to  determine  the  most 
economical  load  for  any  line,  the  derivative  of  Y  must  be  taken 
with  respect  to  the  load,  kw.     It  is  evident  that,  with  the  equa- 
tion in  the  above  form,  the  result  would  be  kw  =  0,  since  the 
minimum  line  loss  is  obtained  with  no  load.     However  the  line 
must  be  considered  as  a  working  unit.     Hence,  if  the  above 
expression  is  changed  to  represent  the  annual  cost  per  kilowatt 
transmitted,  and  the  minimum  value  of  that  quantity  found,  the 
most  economical  load  for  the  line  will  be  discovered. 
If  Y'  =  the  annual  cost  per  kilowatt  transmitted 
^Ka  +KbA       KckwtCe 
kw  A 

K, 


/a          ,\ 
dkw  ~         \       kw2      J  '       A 

-  (27) 


TRANSMISSION-LINE  PROBLEMS  85 

Numerical  Example.  —  The  values  of  the  constants  must  be 
obtained  to  accord  with  local  conditions.     Kc  is  evaluated  as 
follows  : 
Ye  =  annual  cost  of  energy  loss  per  1,000  ft.  = 


Where    /  =  current  per  wire 

R  =  resistance  per  1,000  ft.,  one  wire, 

R  =  -  -  -r1  —      Where  p  =  resistivity  per  mil  foot 


A. 
kw  X  1,000 

=       E  cos  e 


Where  K<  -          x  "  x  365 


if         in  s        K    -  3'940  x  1Q6 

If  "  "  K°~    (E  cos  ey 

Let  us  assume  for  example  a  22,000-volt  line. 
For  which  Kl  =  39 
12.2 


K3  =1.715 

(2.21  +  147.5CCU\  ,,T,        n      _  cost  of  conduc- 
~  \        ~10^~         /  Ccu  =     tor  per  pound 

=  If  Ccu  =  0.20 


5 

K,  =  11.3 
Ka  =  40.72 

43.9 

^6=   TOT 
Kc  =     11.3 

(From  Eq.  24)      7  =  40.72  +  ^A  +  11.3 


V1  1  Q  V    1  05/^ 
"T^V^         =  160. 
4o.y 


40  q 

^ 


T  ll.O 


(From  Eq.  27)fc«,  -  ^40.72 


86 


ECONOMICS  OF  ELECTRICAL  DISTRIBUTION 


The  foregoing  are  all  simple  equations  in  terms  of  the  load 
carried,  the  cross-sectional  area  of  conductor,  the  equivalent 
hours  and  the  cost  of  energy.  The  latter  two  quantities  are 
more  or  less  inter-related.  A  study  of  energy  cost  will  show 
that  the  cost  per  kilowatt-hour  varies,  among  other  things,  with 
the  load  factor  and  hence  with  the  equivalent  hours.  The  value 
of  tCe  may  then  be  obtained  approximately  for  any  value  of  t 
with  any  type  of  load. 


•O.H 


0.04- 


4000  6000  &000 

Load  in  Kilowatts 


10,000 


FIG.  23. — Annual  cost  per  kilowatt  per  1,000  ft.  of  22, 000- volt  transmission  line 
(single  circuit — tCe  =  .06;'  cos  6  =  .85). 

It  will  be  found  advantageous  for  study  to  plot  curves  of  all 
the  equations  given  above.  If  different  values  of  tCe  are  chosen, 
as  tCe  =  .02,  tCe  =  .04,  tCe  =  .06,  tCe  =  .08  and  tCe  =  .10  covering 
the  desired  range  of  values  and  a  sheet  made  up  for  each,  giving  a 
curve,  plotted  between  annual  cost,  Y,  and  load  kw,  for  each  wire 
size,  the  first  equation  (24)  is  well  displayed;  similarly  Eq.  26. 
For  Eqs.  25  and  27,  a  curve,  plotted  between  A  and  kw  for  each 
desired  value  of  tCe,  may  be  obtained. 

For  a  numerical  example  the  value  of  equivalent  hours  is  taken 


TRANSMISSION-LINE  PROBLEMS 


87 


as  6  and  the  corresponding  value  of  tCe  assumed  as  .06.  The 
curves  for  Y',  the  annual  cost  per  kilowatt  per  1,000  ft.,  are 
plotted  as  shown  in  Fig.  23.  The  curves  for  Y,  the  total  annual 
cost  per  1,000  ft.  are  easily  obtained  if  desired  but  are  not  shown 


0  2000  4000         .  6000  8000  10,000          12,000 

Load  in  Kilowatts 

FIG.  24. — Most  economical   conductor  size   for  22,000-volt  transmission  line. 

here.  These  curves  for  Y'  are  interesting  in  that  they  show  defi- 
nitely the  point  of  minimum  cost  or  the  load  of  greatest  economy 
for  any  given  conductor  size.  Likewise,  they  show,  for  any  load, 
the  most  economical  conductor  and  just  how  much  cheaper  one 
size  is  than  another  for  any  load. 

Figures  24  and  25  are  also  plotted  from  Eqs.  25  and  27,  using 


88 


ECONOMICS  OF  ELECTRICAL  DISTRIBUTION 


various  values  for  tCe.  Figure  24  gives  the  most  economical  con- 
ductor size  for  any  load  and  Fig.  25  the  most  economical  load 
for  any  wire  size.  The  curves  for  tCe  =  .06  might  have  been 
derived  directly  from  Fig.  23,  Fig.  24  being  a  bounding  curve 
for  all  the  curves  shown  and  Fig.  25  the  locus  of  all  the  minimum 
points.  It  is  interesting  to  note  that  the  most  economical  size 
of  wire  for  any  load  is  not  the  same  as  the  size  for  which  that  is 
the  most  economical  load.  The  reason  for  this  is  easily  seen 
from  Fig.  23,  for  while  at  2,500  kw.  No.  0  wire  appears  to 


12,000 


06    «*       9Z  #0        *00         »000 

Conductor  Site  in  Circular  Mils 


FIG.  25. — Most  economical  loading  for  22,000-volt  transmission  line. 

be  most  economical,  with  an  annual  cost  of  .051  per  kilowatt 
per  1,000  ft.,  if  a  No.  0  line  is  loaded  to  3,800  kw.,  the  annual 
cost  will  be  only  .0473  per  kilowatt  per  1,000  ft. 

Use  of  Curves. — By  means  of  such  curves  as  these,  a  number  of 
the  problems  of  secondary  transmission  lines  may  be  readily 
solved.  If  the  size  and  characteristics  of  a  load  are  known,  the 
most  economical  conductor  may  be  found  from  the  proper 
curve.  If  some  other  size  than  the  most  economical  is  used, 
its  additional  cost  is  given  by  the  curves.  If  standard  sizes  are 
to  be  determined,  the  range  of  loads,  and  their  characteristics, 
to  be  handled  may  be  studied  in  connection  with  the  curves 
and  the  most  economical  size  or  sizes  to  fit  the  majority  of  cases 
may  be  chosen.  If  the  problem  is  one  of  changing  conductor 
sizes,  the  load  at  which  the  annual  cost  after  the  change  (includ- 
ing the  annual  charges  on  the  cost  of  changes)  will  be  less  than 
the  annual  cost  with  the  present  size,  may  be  taken  from  such 


TRANSMISSION-LINE  PROBLEMS  89 

curves  as  Fig.  23.  Similarly,  if  curves  for  two  (or  more)  circuit 
lines  are  prepared,  the  economy  of  adding  a  second  circuit,  of 
building  an  additional  line,  etc.,  may  be  studied.  (This  problem 
usually  involves  considerations  of  reconstruction,  see  Chap.  X.) 
If  several  lines  of  different  sizes  are  feeding  one  station,  the 
economical  division  of  load  is  shown  by  curves  similar  to  Fig.  25. 

Most  Economical  Route. — Another  problem  which  is  encoun- 
tered in  the  design  of  nearly  all  transmission  lines  is  the  choice 
of  the  best  route.  It  is  seldom  possible  to  follow  anything  like  a 
direct  "air  line"  route  on  account  of  the  difficulty  in  obtaining 
right-of-way.  If  wooden  poles  or  steel  structures  with  small 
base  are  used,  a  route  is  ordinarily  chosen  which  follows  the 
highway  in  such  a  way  as  to  arrive  at  the  destination  with  the 
least  possible  length  of  line  and  the  fewest  possible  difficulties  in 
construction.  It  often  occurs,  however,  that  this  choice  is  not  a 
simple  matter.  There  may  be  two  or  more  routes,  each  of  which 
has  advantages  and  disadvantages  which  more  or  less  offset  each 
other,  and  there  is  no  self  evident  choice  between  them.  It  is 
then  necessary  to  make  a  careful  cost  analysis  to  determine  the 
most  economical  route.  The  relative  economy  of  the  various 
routes  can  be  weighed  against  any  other  features  which  are  not 
subject  to  a  tangible  cost  comparison,  and  the  most  practicable 
route  will  usually  be  made  evident. 

Probably  one  of  the  most  usual  and  simple  problems  of  this 
kind  is  the  choice  between  two  routes,  one  of  which  is  shorter 
than  the  other  but  necessitates  the  purchase  of  a  certain  amount 
of  right-of-way,  while  the  other  is  longer  but  follows  the  highway 
and  is  not  subject  to  right-of-way  charges.  It  is  the  purpose 
here  to  show  how  the  cost  of  two  such  routes  can  be  readily 
compared  and  a  general  equation  derived  which  will  cover  all 
such  problems  on  any  particular  system.  This  equation  can  be 
used  to  determine  curves  that  give  a  more  tangible  method  of 
choosing  the  best  solution. 

As  a  corollary  to  this  problem  we  have  the  condition  where 
there  is  a  choice  of  a  longer  route,  clear  of  trees  and  other  obstruc- 
tions, and  a  shorter  one  which  will  require  extra  high  structures, 
additional  guying,  etc.  In  fact  the  problem  resolves  itself  into 
the  question  of  "How  much  can  we  afford  to  spend  in  addition  to 
the  normal  cost  of  construction  in  order  to  shorten  a  route?" 

The  basis  for  such  a  cost  comparison  will  naturally  be  the 
annual  cost  of  the  two  routes.  If  we  can  determine  what  will  be 


90  ECONOMICS  OF  ELECTRICAL  DISTRIBUTION 

the  saving  in  annual  cost  of  the  shorter  route  over  the  longer 
when  only  normal  line  construction  cost  is  considered,  we  can 
then  tell  what  is  the  maximum  amount  which  we  would  be 
justified  in  expending  for  right-of-way  or  additional  material, 
etc.,  in  order  to  use  that  shorter  route.  If  the  longer  route  should 
also  require  some  extra  construction  expenses  these  should  natu- 
rally be  included  in  the  comparison. 

The  annual  cost  of  any  route  is  made  up  of  the  annual  charges 
against  the  construction  itself  and  the  annual  charges  for  the 
energy  loss,  the  latter  depending  on  the  load  carried  and  the  size 
of  wire  used.  It  is  first  necessary  to  determine  as  accurately  as 
possible  what  the  unit  annual  charges  will  be  on  a  normal  line, 
i.e.  one  without  exceptional  difficulties. 

Determination  of  Normal  Line  Cost. — The  annual  charges 
against  the  construction  will  depend  upon  the  standard  type  of 
construction  used  and  on  the  cost  of  materials  and  labor  in  the 
locality  under  consideration.  There  must  be  obtained  the  total 
cost  per  mile  of  the  line  in  place  including  both  material  and  labor 
costs  on  poles,  crossarms,  pins,  insulators,  wire,  grounds,  guying, 
etc.  In  doing  this,  the  sizes  and  quantities  of  the  various 
materials  which  would  constitute  an  average  normal  mile  of  line 
must  first  be  determined.  The  annual  cost  may  then  be  obtained 
by  figuring  interest,  taxes  and  depreciation  on  this  first  cost 
allowing  for  the  difference  in  depreciation  between  the  different 
materials  used.  A  certain  amount  per  year  .should  also  be  added 
for  maintenance,  patrolling,  etc.  The  resulting  annual  cost  will 
be  a  constant  for  any  condition  of  loading,  providing  the  size  of 
wire  is  fixed,  and  may  be  represented  by  K&. 

Usually  when  a  shorter  route  is  selected  a  number  of  corners 
are  eliminated.  The  cost  of  a  corner  construction,  especially  on 
high-tension  lines,  may  be  quite  an  appreciable  addition  to  the 
normal  cost  of  the  line.  It  can  be  figured  however  in  the  same 
manner  as  the  normal  line  cost.  This  annual  cost  per  corner 
may  be  represented  by  K8. 

Annual  Cost  of  Energy  Losses. — The  annual  charges  due  to 
loss  of  energy  may  be  determined  as  shown  above  in  the  study  of 
conductor  economy. 
Then, 

Annual  cost  of  energy  loss  =  K7kw2"  per  mile. 

,_,         v       365,000  rtCe 
Where  X7=- 


TRANSMISSION-LINE  PROBLEMS  91 

Where  r  =  resistance  of  conductor  in  ohms  per  mile. 

When  these  constants  have  been  evaluated  for  the  particular 
conditions  of  the  problem  in  hand,  it  is  then  possible  to  determine 
the  total  annual  cost  per  mile  with  normal  line  construction  and 
the  total  annual  cost  for  any  particular  length  including  the 
corners. 

Determination  of  General  Equation. — If  we  let  L  represent  the 
difference  in  length  between  the  shorter  route  and  the  longer, 
the  savings  in  annual  cost  of  the  shorter  route  over  the  longer 
would  be  L  times  the  annual  cost  per  mile  if  normal  line  construc- 
tion only  is  considered.  This,  then,  represents  the  maximum 
amount  which  it  would  be  economical  to  spend  in  addition  to  the 
normal  cost  of  the  line  in  order  to  utilize  the  shorter  route.  If 
the  additional  cost  is  for  right-of-way,  this  annual  amount  is  the 
maximum  rent  which  we  could  afford  to  pay  for  it  or  if  the  right- 
of-way  is  bought  outright,  this  amount  represents  the  interest 
and  taxes  on  the  maximum  amount  which  could  be  paid  for  the 
necessary  land.  In  the  case  where  the  additional  cost  would  be 
for  extra  poles  and  other  line  material,  the  above  savings  would 
represent  the  interest,  taxes,  and  depreciation  on  the  maximum 
amounts  which  could  be  so  expended. 

If  L  =  the  length  in  miles  which  would  be  saved  by 

using  the  shorter  route, 

Ci  =  the   maximum   extra   expenditure   allowable   in 
order  to  use  the  shorter  route, 

C 

C  =  the  extra  expenditure  per  mile  saved  =  y*> 

N  =  the  number  of  corners  saved  by  using  the  shorter 

route, 
g  =  per  cent  annual  charges  on  additional  expenditure 

(interest,  taxes,  depreciation), 
KG  =  the  annual  cost  of  normal  line  construction  per 

mile, 
K7kw2  =  annual  cost  of  energy  loss  per  mile, 

KS  =  annual  cost  of  a  corner  in  addition  to  normal  line 

cost, 
Then 

The  annual  charges  on  the  additional  expenditure  for  right-of- 
way,  extra  poles,  etc.  would  be  gd 

gCi  =  L(K6  +  K7kw*)  +  KBN  (28) 


92  ECONOMICS  OF  ELECTRICAL  DISTRIBUTION 

which  is  a  general  equation  and  can  be  applied  to  any  such  case 
if  the  proper  values  for  the  various  constants  are  obtained.  If 
then  the  contemplated  shorter  route  can  be  used  with  a  less 
expenditure  than  C\  it  would  be  an  economical  proposition  to  use 
it. 

The  above  equation  can  be  made  even  more  general  and  be 
shown  graphically  by  means  of  a  curve  if  we  reduce  it  to  terms  of 
C  or  the  maximum  allowable  expenditure  per  mile  of  length  saved, 
omitting  for  the  present  the  amount  saved  on  corners. 


r_       &          7w  (     . 

~17i6o~ 

The  value  of  g  as  indicated  above  depends  upon  the  nature  of 
the  additional  expenditure.  If  it  is  for  right-of-way,  interest  and 
taxes  must  be  charged  but  no  depreciation  need  be  considered. 
We  may  take  interest  at  6  per  cent.,  taxes  at  2  per  cent  or  g  — 
8  per  cent.  This  represents  yearly  rental.  If  extra  line  material 
is  to  be  purchased  depreciation  also  must  be  added.  Considering 
the  life  of  such  material  as  20  years,  depreciation  =  5  per  cent 
and  g  =  13  per  cent. 

Data  for  Curves.  —  The  accompanying  curves,  Fig.  26,  were 
plotted  assuming  values  for  KG,  K7  and  K8  as  follows: 

K6  —  $350  (wood-pole  construction) 
r  =  .  539  for  No.  0  wire 
t  =  6  hr. 
E  =  46,000  volts 
cos  0  =  .90 

Ce  =  .01  per  kilowatt-hour 
365,000  rtCe        0.69 


7 


(E  cos  BY          105     ' 
8  =  $40 
g  =  8  and  13 

.69  kw*  .69  kw* 

"  -- 


C  =  -  T^B  -  and        C  = 


.08  .13 

These  equations  are  plotted  for  various  values  of  the  load,  kw. 
The  lower  curve  A  is  plotted  for  g  =  8  per  cent  or  when  the 
additional  expenditures  to  be  made  in  using  the  shorter  route  will 
be  for  such  property  as  right-of-way  which  has  no  depreciation. 


TRANSMISSION-LINE  PROBLEMS 


93 


In  case  the  right-of-way  is  obtained  on  a  yearly  rental  basis, 
8  per  cent  of  the  amount  shown  by  the  curve  would  be  the  allow- 
able yearly  rent.  The  upper  curve  is  to  be  used  when  the  extra 
cost  will  be  for  extra  line  material  such  as  poles,  stubs,  guys,  etc. 
for  which  the  life  was  assumed  to  be  20  years  and  g  =  13  per  cent. 
If  any  corners  will  be  saved  by  using  the  shorter  route,  the  cost 
of  these  may  be  added  to  the  amount  shown  by  the  curve  or 


10,000 
8000 
6000 
4000 

200O 

/ 

/ 

/ 

/ 

/" 

/ 

2 

/ 

/ 

/A  -  When  expenditure  is  for) 
having  no  depreciation 

B  -  When  expenditure  is  fbrpr 
having  20  year  life 

property 
vperry 

/ 

/ 

If  arty  corner 
shown  by  at 
for  each  co> 

s  are  elimina 
rve^  $500  fot 
'ner 

ted}ae(dto& 
-A}  $308  fo 

mount- 
rB; 

o 

/ 

'2000  4000  6000  8000  10,000          12,000          14,000 

Maximum  Economical  Expenditure  per  Mile  Saved,  dollars 

FIG.   26. — Curves   showing  maximum   economical   expenditure  in   addition   to 
normal  line  cost  in  order  to  shorten  a  route. 


TTOO  ^or  eac^  corner-     ^  ^8  =  $40,  this  amounts  to  $500  for 
g  =  8  per  cent  and  $308  for  g  =  13  per  cent. 

Discussion  of  Curves. — These  curves,  then,  show  the  amount 
which  could  be  spent  in  constructing  a  line  in  addition  to  the 
normal  cost  of  construction  under  average  conditions  in  order  to 
shorten  the  length  of  the  route  1  mile.  The  value  of  C  when  the 
load  is  0  represents  annual  charges  on  the  normal  construction 
cost  of  1  mile  of  line  capitalized  at  the  percentage  g,  which  is  the 
percentage  of  annual  charge  against  the  additional  property 


94  ECONOMICS  OF  ELECTRICAL  DISTRIBUTION 

acquired  and  depends  on  its  nature.  As  the  load  increases  the 
value  of  C  increases  due  to  the  increasing  cost  of  the  line  loss. 

The  slope  of  the  line  away  from  the  vertical  illustrates  graph- 
ically the  importance  of  knowing  as  accurately  as  possible  the  load 
and  the  load  factor  that  it  is  intended  to  carry  on  the  line,  in 
order  to  correctly  make  the  choice  of  routes.  Though  an  accu- 
rate determination  may  be  practically  impossible,  it  is  usually 
possible  to  work  between  some  assumed  limits  that  will  represent 
a  wide  range  of  conditions  and  still  lend  themselves  to  analysis 
by  means  of  the  curves. 

Furthermore,  as  the  load  increases,  the  point  which  stands  out 
plainly  is  the  great  economy  which  will  ordinarily  be  achieved 
by  the  use  of  a  shorter  route.  For  example  if  right-of-way  can 
be  obtained  for  a  yearly  rental  of  $1  per  pole,  at  8  per  cent  this 
would  represent  $12.50  per  pole  or  at  35  poles  per  mile,  $437.50 
per  mile.  With  a  load  of  4,000  kw.  the  curves  show  that  we 
could  spend,  according  to  curve  A,  $5,300  for  each  mile  saved, 
which  would  purchase  right-of-way,  at  the  above  rate  for  12 
miles.  Or,  from  curve  B  we  could  expend  $3,300  for  additional 
material  in  order  to  save  1  mile.  If  any  corners  were  eliminated 
the  amount  would  be  still  more.  Hence,  if  the  actual  expenditure 
necessary  is  any  less  than  the  above  figures,  the  difference  would 
represent  a  real  saving  and  it  would  be  economy  to  use  the 
shorter  route. 

Naturally  the  curves  here  given  could  not  be  used  for  any  but 
the  particular  type  of  construction  and  voltage  for  which  they 
were  computed.  It  is  a  simple  matter,  however,  to  develop 
similar  curves  to  fit  any  other  case  from  the  equations  given. 
The  economy  shown  by  such  a  curve,  however,  cannot  be  used 
as  an  absolute  criterion  in  the  choice  of  a  route.  There  are  other 
factors  which  do  not  lend  themselves  to  such  an  exact  cost 
analysis.  The  matter  of  patrolling  a  line  and  making  repairs 
might  be  considerably  more  expensive  along  private  right-of-way 
than  along  a  main  highway.  The  proximity  to  a  railroad  might 
considerably  affect  the  cost  of  erection.  The  matter  of  protec- 
tion against  severe  storms  must  be  considered.  Many  other 
details  will,  in  any  construction  project,  present  themselves  for 
analysis.  If,  however,  we  have  a  concrete  comparison  of  the 
relative  economy  of  one  route  over  another,  we  have  gone  a  long 
way  toward  an  exact  determination  of  which  will  be  the  most 
advantageous  under  all  conditions. 


TRANSMISSION-LINE  PROBLEMS  95 

As  brought  out  at  the  beginning  of  this  discussion,  only  one 
specific  point  in  economies  of  routes  has  been  covered  here,  that 
is,  where  the  choice  between  two  routes  is  to  be  made,  the  shorter 
one  requiring  purchase  of  right-of-way  or  installation  of  higher 
structures  and  special  reinforcements.  The  many  other  prob- 
lems in  the  choosing  of  the  economical  route  for  a  transmission 
line  could  be  treated  in  a  very  similar  manner.  The  question  of 
the  economy  of  diverging  from  the  shortest  route  in  order  to 
utilize  old  poles  already  in  place  is  taken  up  in  the  next  chapter 
under  "  Reconstruction  Problems." 

The  above  will  give  an  idea  of  the  kind  of  problems  in  con- 
nection with  a  secondary  transmission  system  which  require  a 
solution  for  economy.  It  must  always  be  kept  in  mind,  of 
course,  that  economy  is  not  the  only  consideration  in  designing 
a  line.  Mechanical  strength  is  an  important  feature.  Good 
regulation  is  essential.  While  the  economical  conductor  size  is 
usually  independent  of  the  length  of  the  line,  regulation  depends 
on  the  length.  Often,  on  a  long  line,  a  larger  size  than  the  most 
economical  must  be  used  to  give  good  voltage.  In  case  artificial 
means  of  improving  regulation  are  considered,  their  cost  will 
tend  to  offset  the  economy  of  the  smaller  conductor  and  a  study 
of  the  line  as  a  whole  including  all  such  items  is  necessary.  The 
problem  is  still  one  of  economy.  In  any  case  the  determination 
of  the  most  economical  conductor  size,  voltage,  span,  route,  etc. 
will  serve  as  a  starting  point  for  the  study  of  the  most 
advantageous  design  for  final  adoption. 


CHAPTER  X 
RECONSTRUCTION  PROBLEMS 

PRINCIPLES  INVOLVED  IN  THE  SOLUTION  OF  PROBLEMS  DEALING 

WITH  THE  ALTERATION  OR  RECONSTRUCTION  OF  LINES 

ALREADY  BUILT  — METHOD  OF  INCLUDING  VALUE 

OF  SALVAGED   MATERIAL   IN   COST   STUDY 

A  great  number  of  the  problems  in  transmission  and  dis- 
tribution lines,  involve  the  consideration  of  reconstruction,  that  is 
the  improvement  or  enlargement  of  a  system  already  in  oper- 
ation. For  example,  it  may  be  desired  to  determine  the  advis- 
ability of  shortening  a  line  already  in  operation  by  rebuilding 
part  of  it  over  a  new  route.  Or,  it  may  be  necessary  to  meet  an 
unforseen  increase  in  the  load  carried,  by  rebuilding  an  old  line, 
increasing  the  size  of  the  wire,  adding  an  additional  line  on  the 
same  poles,  or  changing  the  voltage.  While  this  chapter  deals 
primarily  with  transmission  line  reconstruction,  the  principles 
enumerated  may  be  applied  to  such  problems  on  any  part  of  the 
system. 

For  convenience,  the  term  "reconstructed  line"  will  be  used  to 
designate  the  line  after  the  change  is  made.  In  any  such  case, 
it  is  necessary  to  consider  the  fact  that  the  old  line  has  more  or 
less  of  its  serviceable  life  left  and  the  value  of  this  must  be  added 
to  the  cost  of  the  new  construction  in  determining  the  total 
investment  represented  by  the  finished  line.  This  total  is  the 
amount  upon  which  the  reconstructed  line  must  pay  a  return 
and  should  be  justified  by  the  increased  capacity  secured.  Or, 
in  a  problem  of  economy,  such  as  shortening  a  line,  it  is  the 
amount  upon  which  the  annual  cost  of  the  new  installation  must 
be  figured.  This  annual  cost  must  be  less  than  that  of  the  old 
line  if  there  is  to  be  any  advantage  in  the  change. 

Total  Investment  Represented  in  Reconstructed  Line. — In 
order  to  determine  the  total  annual  cost  of  the  reconstructed 
line  or  section  of  line,  it  is  easier  to  first  consider  the  factors  which 
affect  the  total  investment  involved,  either  to  increase  or  decrease 
it.  It  will  at  first  be  assumed  that  the  new  line  is  to  be  entirely 
of  new  construction  and  the  old  line  is  to  be  salvaged.  Later, 
the  use  of  old  material  in  the  new  line  will  be  discussed.  It  is 
evident  that,  for  the  purpose  of  this  analysis,  the  amount  of 
investment  represented  by  the  old  line  as  it  stands  must  be 
computed  on  the  principle  of  the  cost  to  reproduce  it  in  its 

96 


RECONSTRUCTION  PROBLEMS  97 

present  condition,  at  present  prices.  It  is  to  be  replaced  by  new 
construction  at  present  prices  and  the  comparison  of  annual  costs 
of  old  and  new  must  be  on  the  same  basis.  This  amount  may 
be  obtained  by  figuring  the  detailed  cost  of  such  a  line,  including 
material  and  labor,  at  present  prices  and  subtracting  from  this  a 
percentage  of  the  whole  equal  to  the  fraction  of  the  total  assumed 
life  of  the  line  which  has  already  elapsed.  The  value  of  the 
elapsed  life  is  assumed  to  have  been  paid  for  in  the  return  from 
the  operation  of  the  line  up  to  the  present  time.  If,  therefore, 
a  new  line  is  built  to  replace  the  old  and  the  old  line  is  removed, 
there  is  added  to  the  total  investment  the  cost  of  all  the  new 
material  used  and  the  labor  involved  in  the  change.  There  is 
subtracted  from  the  total  investment  the  salvage  value  of  the 
material  recovered.  The  investment  represented  by  the  old 
line  as  it  stands  is  thus  increased  by: 

1.  The  cost  of  the  material  used  in  the  reconstructed  line. 

2.  The  cost  of  the  labor  necessary  in  the  building  of  the 
reconstructed  line. 

3.  The  cost  of  the  labor  necessary  in  dismantling  and  removing 
to  the  warehouse  the  material  in  the  old  line. 

The  investment  is  decreased  by: 

The  present  value  of  the  material  in  the  old  line,  (i.e.,  its  cost 
if  new,  less  a  percentage  of  depreciation  for  age). 
Hence,  the 

Net  increase  in  investment  =  Cost  of  material  in  reconstructed  line 
+  Cost  of  labor  in  reconstructed  line 
+  Cost  of  labor  removing  old  line 
—  Present  value  of  material  in  old  line. 
As  was  explained  above, 

Investment   represented   by   the    old   line  =  Present   value    of 

material  in  old  line 
+  Present  value  of 
labor  in  old  line. 

It  is  evident  that  the  last  item  in  "net  increase  in  investment" 
cancels  the  first  item  in  "  investment  represented  by  the  old 
line,"  hence  this  quantity  "  present  value  of  material  in  old 
line"  does  not  enter  into  the  " total  investment  represented." 
Therefore, 
Total  investment  .represented  =  Present  value  of  labor  in  old  line 

by  the  new  line  +  Cost  of  material  in  reconstructed  line 

-|-  Cost  of  labor  in  reconstructed  line  (30) 
+  Cost  of  labor  removing  old  line. 


98  ECONOMICS  OF  ELECTRICAL  DISTRIBUTION 

In  other  words,  a  new  line,  built  to  replace  an  old  one  which 
has  not  outlived  its  usefulness,  represents  an  investment  including 
the  present  value  of  the  labor  of  erecting  the  old  line  and  the  cost 
of  labor  in  removing  the  old  line  in  addition  to  the  total  cost  of 
material  and  labor  in  erecting  the  new  line. 

Annual  Cost  of  Reconstructed  Line. — The  subject  of  annual 
cost  in  general  has  been  discussed  in  Chap.  Ill  and  the  forma- 
tion of  the  general  equation  in  Chap.  VI.  The  application  to  the 
special  problem  of  the  reconstructed  line  will  be  taken  up  here. 

Investment  Charges. — In  order  to  obtain  the  yearly  cost  charge- 
able to  the  reconstructed  line  which  is  the  quantity  which  must 
be  used  in  considering  its  economy,  it  is  necessary  to  spread  this 
total  investment  over  a  period  of  time  estimated  for  the  new 
construction.  The  percentage  to  be  charged  annually  must 
include  interest,  taxes,  insurance  and  depreciation.  Ordinarily, 
at  least  two  different  percentages  will  be  used  in  actual  computa- 
tion. One  will  apply  to  labor  investment  which  has  no  salvage 
value,  and  to  materials  such  as  poles,  crossarms,  pins,  etc.,  which 
have  a  comparatively  short  life,  and  whose  scrap  value  at  the 
end  of  the  useful  life  is  probably  about  equal  to  the  cost  of  labor 
necessary  for  salvaging.  Another  figure  will  be  used  on  such 
material  as  bare  copper  wire  which  has  practically  no  physical 
depreciation.  Its  value  at  the  end  of  the  life  of  the  line  is  as 
great  as  at  present,  providing  prices  do  not  decrease,  and  the 
only  expense  incurred  will  be  the  labor  cost  of  salvaging.  The 
market  value  of  copper  has  varied  so  widely  during  the  past  few 
years  that  a  great  element  of  uncertainty  is  introduced  into  this 
item.  For  this  discussion,  however,  it  will  be  assumed  that  the 
price  of  copper  will  remain  constant.  For  the  present  both  these 
percentages  will  be  represented  by  the  letter  g. 

g  =  per  cent  interest,  taxes,  insurance  and  depreciation 
chargeable  annually  against  the  construction.  The  expression 
11  g  X  any  quantity"  merely  indicates  that  it  is  the  annual  charge 
against  that  quantity  which  is  being  considered. 

Operating  Charges. — Another  element  of  annual  cost  is  the 
operating  expense.  The  chief  item  under  this  head  will  be  that 
of  energy  loss.  If  the  reconstructed  line  is  of  higher  voltage  or 
of  larger  wire  size  than  the  old,  or  if  the  length  is  shortened, 
the  energy  loss  may  be  materially  reduced.  This  will  tend  to 
reduce  the  annual  cost  and  is  an  important  item  when  relative 
economy  is  to  be  considered.  Here,  again,  considerable  uncer- 


RECONSTRUCTION  PROBLEMS  99 

tainty  is  introduced  on  account  of  the  variation  in  load  factor 
and  load  at  present  and  in  the  future.  The  value  of  these  items 
must  be  selected  for  the  particular  problem  by  careful  study 
of  the  local  conditions,  since  here,  as  in  any  engineering  problem, 
the  solution  must  be  based  on  certain  definite  assumed  values 
for  all  variable  quantities.  The  success  or  failure  of  the  solution 
will  depend  upon  the  good  judgment  used  in  such  selection. 
The  operating  expense  also  includes  the  cost  of  superintending, 
repairing,  patrolling,  etc.,  which  will,  in  most  cases,  probably 
not  be  materially  different  on  the  reconstructed  line  from  what 
it  was  on  the  old  line.  In  case  the  difference  is  marked  this  point 
must  be  taken  into  consideration  in  the  final  decision. 

General  Equation  for  Annual  Cost  of  Reconstructed  Line. — 
A  general  equation  may  now  be  formulated  for  the  annual  cost 
of  the  reconstructed  line  by  combining  these  factors. 


Annual  cost  of  ^esen*  value  .°*  labor  in  °ld  line:  v 

reconstructed  +Cost  of  material  in  reconstructed  line. 

+Cost  of  labor  in  reconstructed  line.  i  ,„.,,. 

+Cost  of  labor  removing  old  line. 


line  =  9x 


+ Cost  of  superintendence,  repairs,  patrolling,  etc. 
+Cost  of  energy  losss  on  recontructed  line. 
Similarly, 

Value  of  material  in  old  line. 


The  annual  cost  on 
the  old  line  =  gX 


+ Value  of  labor  in  old  line. 

+Cost    of    superintendence,    repairs, 

patrolling,  etc. 
+Cost  of  energy  loss  on  old  line. 


(32) 


By  proper  selection  and  application  of  the  percentage  g,  and 
computation  of  the  various  costs  involved,  in  accordance  with 
local  conditions,  these  equations  may  be  applied  to  any  such 
problems  of  reconstruction  such  as  determining  the  relative 
economy  of  replacing  a  present  line  with  one  of  higher  voltage 
or  larger  wire,  or  of  shortening  a  line  by  rebuilding  part  of  it,  as 
will  be  shown  later. 

Utilization  of  Old  Materials  in  Reconstructed  Line. — In  the 
above  discussion  it  was  assumed  that  all  the  old  material  would 
be  salvaged.  It  very  often  occurs,  however,  in  such  cases  as  an 
increase  in  voltage,  that  the  new  line  may  to  advantage  follow 
the  old  route  for  part  of  the  distance  at  least,  using  the  old  poles 
and,  in  some  cases,  the  old  crossarms,  insulators,  etc.  The 
general  equation  still  holds  good  for  this  condition  if  its  deriva- 


100  ECONOMICS  OF  ELECTRICAL  DISTRIBUTION 

tion  is  kept  in  mind  and  the  various  items  adjusted  to  fit  the  con- 
ditions. The  "cost  of  material  in  reconstructed  line"  must 
include  the  " present  value"  of  the  old  material  used,  since,  it 
will  be  remembered,  this  originally  entered  the  equation  as  a 
credit  (to  be  salvaged)  and  was  cancelled  out.  The  "cost  of 
labor  in  reconstructed  line"  must  be  reduced  due  to  the  fact 
that  the  old  material  used  is  already  in  place  but  it  must  also 
include  any  labor  cost  necessary  to  adapt  the  old  material  to  the 
new  conditions.  The  "cost  of  labor  of  removing  old  line"  will, 
of  course,  be  reduced  by  the  amount  that  would  be  required 
for  the  material  not  removed.  The  value  of  g  must  also  be 
selected  to  take  into  account  the  fact  that  the  present  life  of  the 
old  material  will  not  be  as  great  as  the  new  and  hence  the  labor 
necessary  in  adapting  it  will  have  a  greater  yearly  depreciation. 
With  these  conditions  clearly  in  mind,  however,  it  is  evident 
that  the  general  equation  may  be  safely  applied  to  any  such 
problem  of  reconstruction,  even  when  part  of  the  old  construc- 
tion is  to  be  utilized.  This  would  include  also  the  case  when  a 
second  line  was  to  be  added  on  the  same  poles. 

So  far  the  discussion  has  been  entirely  of  a  general  nature.  It 
has  been  shown,  how,  in  any  problems  involving  reconstruction 
of  a  line  still  serviceable,  the  investment  cost  and  annual  charges 
pertaining  to  the  old  line  must  be  included  in  the  costs  chargeable 
to  the  new  line.  These  costs  will  be  increased  or  decreased  by  the 
various  extra  expenditures  or  savings  belonging  to  the  new  line. 
It  is  now  proposed  to  show  how  the  general  method  can  be  applied 
to  specific  cases,  and  how  in  some  cases  "short-cuts"  may  be 
introduced  to  greatly  simplify  the  problem. 

Economy  of  Shortening  a  Line  already  in  Service. — Let  us 
assume,  for  example,  a  line  several  years  old  which  is  in  good 
condition  and  of  sufficient  capacity  to  care  for  the  predicted  load 
up  to  the  probable  length  of  its  life.  Since  its  construction,  how- 
ever, conditions  have  so  changed  that  it  is  now  possible  to  reduce 
its  length  considerably  by  using  a  more  direct  route  in  some  places. 
The  question  arises  as  to  whether  or  not  it  would  be  economical 
to  rebuild  those  portions  of  the  line  by  the  shorter  routes.  In 
order  to  determine  this  it  is  necessary  to  compare  the  annual 
cost  of  the  present  installation  with  the  total  annual  cost  charge- 
able to  the  reconstructed  installation,  if  the  change  is  made. 
These  annual  costs  may  be  determined  by  use  of  the  general 
equations  previously  developed. 


RECONSTRUCTION  PROBLEMS  101 

The  present  annual  cost  charged  to  the  section  of  the  line  under 
consideration  is  given  by  Eq.  32. 

The  annual  cost  in  case  the  old  line  is  replaced  by  the  new 
section  would  be  as  shown  by  Eq.  31. 

It  would  be  economical,  then,  to  undertake  the  new  construc- 
tion only  if  the  annual  cost  of  the  reconstructed  line,  thus  com- 
puted, is  less  than  that  of  the  old  line.  The  added  construction 
cost  must  be  offset  by  the  reduced  energy  loss.  The  limiting 
case  is  when  the  two  annual  costs  are  equal: 

Annual  cost  of-  reconstructed  line  =  Annual  cost  of  old  line. 
It  is  evident,  however,  that  several  of  the  items  in  both  annual 
costs  are  practically  the  same,  so  for  the  purpose  of  this  compari- 
son they  may  be  eliminated.  Since  the  reconstructed  section  is 
but  a  small  part  of  the  line  as  a  whole,  its  useful  life  must  be 
assumed  to  be  that  remaining  in  the  old  line.  Hence  the  annual 
charges  against  the  "  present  value  of  labor  in  old  line,"  which  is 
part  of  the  "annual  cost  of  the  reconstructed  line,"  will  be  prac- 
tically equal  to  the  annual  charges  against  "cost  of  labor  in  old 
line,"  which  is  part  of  the  "annual  cost  of  old  line."  The  cost 
of  "superintendence,  repairs,  patrolling,  etc.,"  will  be  practically 
the  same  for  both,  unless  the  difference  in  length  is  large.  The 
equation  then  reduces  to 

Cost  of  material  in  recon- 


structed  line  + 
Cost    of   labor   in  recon- 
structed line  + 
Cost    of    labor  removing 


g  (Cost   of   material  in  old 

line) 

+  Cost  of  energy  loss  on  old 

line  (33) 


old  line 
+  Cost  of  energy  loss  on  re- 
constructed line 

The  cost  of  material  and  labor  in  the  new  line  will  of  course 
include  the  cost  of  any  private  right-of-way  which  it  is  necessary 
to  purchase  and  of  any  other  extra  expense  connected  with  the 
construction. 

A  system  of  symbols  will  be  adopted,  conforming  to  those 
previously  used. 

Let  L2  =  length  in  miles  of  the  new  section  to  be  built, 
Le  =  length  in  miles  of  the  old  section  to  be  replaced, 
kw  =  load  in  kilowatts, 

L4  =  the  length  in  miles  of  right-of-way  to  be  purchased, 
N  =  number  of  new  corners  necessary  in  reconstructed  line, 
Cr  =  cost  per  mile  for  right-of-way. 


102  ECONOMICS  OF  ELECTRICAL  DISTRIBUTION 

The  details  of  the  computation  of  unit  costs  used  here  need  not 
be  given  as  they  would  apply  to  only  one  particular  case.  The 
figures  used  are  based  on  a  line  at  46,000  volts,  single  construction, 
on  wooden  poles,  with  No.  0  bare  copper  wire,  energy  costing 
1  ct.  (.01)  per  kilowatt-hour.  The  annual  cost  on  such  new 
construction  introduced  into  an  old  line  will  vary  somewhat  with 
the  age  of  the  line  as  explained  above.  However,  since  such  a 
reconstruction  would  probably  not  be  considered  except  during 
the  early  life  of  the  line,  and  of  course,  the  shorter  the  remaining 
useful  life  assumed,  the  greater  the  salvage  value  of  the  new 
material  at  the  end  of  that  life,  this  may  be  assumed  to  be 
constant  for  any  age.  This  assumption  would  also  be  favored 
by  the  probability  that  this  new  section  would  be  used,  at  least 
in  part,  beyond  the  life  of  the  rest  of  the  line. 

The  unit  costs  are  then  assumed  to  be  as  follows  : 

Annual  charges  on  material  and  labor  in  recon- 

structed line  =  $360  .  00  per  mile 

Annual  charges  on  cost  of  labor  removing  old  line        =      30  .  00  per  mile 
Annual  charges  on  cost  of  material  on  old  line  =    225  .  00  per  mile 

Annual  cost  of  energy  loss  =        0  .  69  kw2  per  mile 

105 

Annual  cost  of  new  corner  =      40.00 

Annual  cost  of  right-of-way  =        0  .  08  Cr  per  mile 

Substituting  these  values,  the  equation  for  determining  economy  becomes 

360L2  +  40AT  +  0.08CrL4  +  30L6  +  °kw2L2  =  225L6 


165L2  +  40N  +  0.08CrL4  =  (L6  -  L2)    jg*™2    +    19$         (34) 

There  are  too  many  variables  to  exhibit  this  equation  as  a  curve. 
The  limiting  values,  however,  can  be  so  expressed.  If  no  right- 
of-way  need  be  purchased  and  no  new  corners  are  added,  the 
greatest  length  of  a  new  line,  L2,  in  order  to  effect  a  saving  of 
1  mile  (Le  —  L2  =  1)  would  be 


L^L,  -         —65-  ~W~kW     +   L18 

This  is  plotted  in  curve  A  (Fig.  26  a)  .  This  curve  shows  the  great- 
est length  of  new  construction  which  could  be  economically  built 
under  the  most  favorable  conditions,  that  is,  normal  line  con- 
struction cost  with  no  corners  and  no  other  extra  expense  for 
right-of-way,  etc.,  for  each  mile  subtracted  thereby  from  the 


RECONSTRUCTION  PROBLEMS 


103 


total  length  of  the  line.     If  such  other  extra  expense  is  necessary 

the  value  of  j f-  will  be  less,  as  may  be  seen  from  the  equa- 
te —  Lt2 

tion  (34) .  Hence  this  curve  may  be  used  as  a  test.  If  the  length 
of  new  line  under  consideration  is  greater  than  that  shown  by  the 
curve,  it  will  not  be  economical  to  build  it.  If  it  is  less,  the  exact 
economy  must  be  determined  from  the  equation  (34).  For 


12,000 


10,000 


8000 


4000 


2000 


A  -  New  construction  art  normal  line 
cost  only,  with  no  extra  expense  for 


B  -New  construction  on  private  right, 
of  way  at  $35  per  mile  per  year 

If  'extra  corners  are  necessary^ubtracf 
as  shown  ty  curve 


from  value 

0.2 


f°r  each  corner 


(  e  3  4  5  6  7 

Length  of  New  Construction  per  Mile  of  Length  Saved=  Jd  Miles 


FIG.  26o. — Curve  showing  maximum  economical  length  of  new  construction, 
replacing  old  line,  per  mile  of  length  eliminated. 

example,  as  a  rather  extreme  case,  suppose  there  are  extra  corners 
necessary  and  the  total  distance,  L2,  will  be  on  private  right-of- 
way  costing  $35  per  mile  per  year. 

T    L*       =  *^kw*  +  0.975  -  T°'2NT  (36) 

*-/6   —   -L/2  -L/6    —  JL/2 

If  the  last  term  is  omitted  this  equation  may  be  plotted  as  shown 
by  the  curve  B,  Fig.  26,  and  gives  the  economical  values  of 

T j-    (new   construction   per   mile   saved)    when   the   extra 

LQ   —  LZ 

0  2 
construction  cost  is  fairly  large.     For  extra  corners  7—  —7-  miles 

Ltfy   —  JL/2 


104  ECONOMICS  OF  ELECTRICAL  DISTRIBUTION 

must  be  subtracted  from  the  value  shown  by  the  curve  for  each 
additional  corner.  Of  course  there  might  be  other  extra  costs 
for  more  poles,  higher  poles,  etc.,  which  have  not  been  especially 
considered  here.  These  could  be  included  in  the  same  manner 
however. 

The  economy  for  any  particular  loading  on  such  a  line  may  now 
be  readily  discovered  from  the  curves  and  equations.  If,  for  ex- 
ample, under  the  conditions  assumed,  the  load  is  6,000  kw.  for  no 
corners  and  no  cost  of  right-of-way,  according  to  curve  A,  the 
new  construction  could  be  built  for  2.7  miles  for  each  mile  saved 
thereby,  or  say,  8.1  miles  in  order  ta  save  3  miles.  In  case  the 
new  construction  requires  private  right-of-way,  as  per  curve  B, 
2.2  miles  may  be  built  for  each  mile  saved  or  6.6  to  save  three 
miles.  If  two  extra  corners  were  necessary,  the  figure  would  be 
1.8  miles  for  only  one  mile  saved,  or  for  three  miles,  2.067  per 
mile  or  6.2  miles  and  so  forth. 

There  is  thus  displayed  a  tangible  method  of  exhibiting  the 
economy  of  shortening  an  old  line  by  reconstruction.  For  any 
particular  system,  of  course,  the  costs  must  be  derived  inde- 
pendently. From  them,  similar  curves  can  be  drawn  and  applied 
to  the  solution  of  such  problems. 

Economy  of  Using  Part  of  Old  Line  Equipment  in  its  Original 
Location  for  Reconstructed  Line. — Another  characteristic  prob- 
lem of  this  general  nature  is  the  following.  In  building  a  new 
line  to  replace  an  old  one,  which  is  not  worn  out,  it  may  be 
possible  to  use  a  considerable  length  of  the  old  line  poles  or  other 
materials  in  place.  However  some  shortening  of  the  length  of 
the  line  might  be  accomplished  by  using  a  more  direct  route  on 
entirely  new  construction.  How  far  from  the  direct  route  may 
the  line  diverge  economically  in  order  to  use  this  old  line  material? 

The  solution  is,  again,  reached  by  a  comparison  of  the  annual 
costs  on  the  two  routes  using  the  general  equation  already 
developed.  This  comparison,  however,  may  be  considerably 
simplified,  as  in  the  preceeding  problem,  by  making  use  of 
certain  short  cuts  which  reduce  the  number  of  terms  to  be  con- 
sidered. The  "present  value  of  labor  in  old  line"  is  a  cost 
chargeable  to  the  reconstructed  line  as  a  whole.  It  will  be  the 
same  no  matter  whether  the  old  material  is  used  in  place  or  not. 
Hence,  it  may  be  omitted  in  the  comparison  of  economy  of  the 
two  methods  of  construction  on  any  section  of  line  where  the  use 
of  old  material  is  considered.  The  "cost  of  labor  removing  the 


RECONSTRUCTION  PROBLEMS  105 

old  line  "  is  also  chargeable  to  the  new  line  as  a  whole  and  it  might 
be  somewhat  difficult  to  properly  apportion  the  amount  to  be 
charged  to  any  particular  section.  This  may  be  avoided  if  it  is 
considered,  for  the  comparative  costs,  that  this  item  is  the  same 
amount  for  both  alternative  constructions  and  hence  may  be 
omitted  from  both.  Since,  however,  some  of  the  old  material 
left  in  place  will  reduce  the  actual  amount  of  this  item  in  the  one 
case,  the  cost  represented  by  the  old  materials  used  must  be 
adjusted  by  subtracting  from  their  present  value  the  amount 
which  it  would  have  cost  to  remove  them.  It  is  seen  that  this 
method  is  equivalent  to  'charging  both  alternatives  with  the 
proper  share  of  the  "cost  of  removing  old  line"  and  then  sub- 
tracting from  one  the  removal  cost  eliminated  by  the  use  of  the 
old  material  in  place.  It  is  also  equivalent  to  considering  that 
old  material  in  place  should  be  charged  at  its  warehouse  value 
which  is  its  present  value  as  material,  if  available,  less  the  cost  of 
making  it  available  or  removing  it  to  the  warehouse.  The 
adjusted  annual  costs  of  the  reconstructed  line  on  old  poles,  and 
of  new  construction,  per  mile,  may  be  thus  obtained  and  are 
convenient  figures  to  use  in  all  such  economic  comparisons.  It 
must  always  be  borne  in  mind  however  that  these  are  adjusted 
costs  and  are  not  the  true  annual  costs  chargeable  to  the  line. 

The  comparison  of  the  annual  costs  of  the  two  routes  may  now 
be  made  by  comparing  the  values  obtained  from  the  expression 

g  (cost  of  material  and  labor  (adjusted  cost))  +  cost  of  energy 
loss; — with  the  proper  selection  of  g,  for  both  alternatives. 

The  following  symbols  have  been  used: 

LI  =  length  in  miles  of  the  old  pole  line  which  can  be  used, 
L2  =  length  in  miles  of  the  alternative  new  line, 
L3  =  length  in  miles  of  the  new  construction  necessary  to  supple- 
ment the  old  pole  line,  i.e.,  bringing  the  line  from  the  new  route 
to  the  old  if  necessary, 

N  =  the  number  of  new  90°  corner  constructions  saved  by 

using  the  more  direct  route, 
kw  =  load  carried  in  kilowatts, 
a  =  age  of  old-pole  line  in  years, 
Ce  =  cost  of  energy  per  kilowatt  hour, 
Z/4  =  length  in  miles,  of  right-of-way  to  be  purchased, 
Cr  =  purchase  price  per  mile  of  right-of-way. 

The  figures  here  given  are  based  on  an  old  line  on  untreated 


106  ECONOMICS  OF  ELECTRICAL  DISTRIBUTION 

wood  poles,  new  line  on  treated  poles,  voltage  of  reconstructed 
line  46,000  volts  necessitating  new  crossarms  and  insulators 
throughout,  wire  No.  0  bare  copper  on  both  old  and  reconstructed 
lines. 

The  annual  charges  obtained  under  these  conditions  are  as 
follows:  Annual  charges  on  new  construction  per 

mile  =  $350.00 
Annual  charges  on  reconstruction  on  old 

,  .,         3,350  -  260a  +  3.2a2 

poles  per  mile  =  -  — — — 

15— a 

Annual  charges  on  energy  loss 

Annual  charges  on  corner  construction     =  $38 . 00 
It  is  now  a  question  of  whether  the  annual  cost  on  LI  -f-  L3  is 
greater  or  less  than  on  L2  using  the  above  figures.     It  would  be 
advantageous  to  divert  to  the  old  pole  line  when 
'3,350  -  260a  +  3.2a2 


,  e 

-a  ~ro~; 

Ls  (350  +  ™g^)  +  38AT/Z,  (350 

.(L2  -  L3)  (350  +  69fr*Ce)  -  387V  +  .08L4CV 
T  /  _  :  _  ir  _  _  ___ 

\  3,350  -  260a  +  3.2a2       69kw*Ce 

15-a  105 

An  example  of  the  use  of  this  equation  would  be  in  such  a  case 
as  that  of  a  line  following  a  private  right-of-way  instead  of  a 
highway  thereby  eliminating  four  corners  and  considerable 
distance.  If  we  assume  the  old  line  to  be  5  years  old  (a  =  5) 
and  Ce  —  .01  and  right-of-way  at  $35  per  mile  per  year  the 
equation  would  become  for  that  particular  case  with  L4  =  L2; 
L3  =  0;  #  =  4. 

(350  +  ~^-2)   -  152  +  35L2 


Ll 


3,350  -  1,300  +  80       .69/cw2 
10  105 


(385  +  ^rgr-J   "  152 

— (38) 


:  +  213 

which  is  easily  solved  for  known  values  of  LI,  L2  and  the  load  and 
the  relative  economy  thus  determined.     If  LI  is  greater  than  the 


RECONSTRUCTION  PROBLEMS 


107 


second  member,  the  economy  evidently  lies  in  the  new  route,  if 
less,  in  the  old  one.  No  curve  could  be  plotted  which  would  be 
of  any  great  value  in  this  instance  since  there  are  so  many 
variables  in  the  expression  which  are  fixed  for  only  one  particular 
problem  or  condition. 

Application  of  Above  Method  of  Choice  of  Transmission 
Route. — There  is  a  very  useful  application  of  the  principles  of 
this  last  problem  in  the  choice  of  a  route  for  a  transmission  line 
as  a  whole.  When  there  are  several  alternative  routes  a  condi- 
tion similar  to  the  following  concrete  example  is  very  often 
encountered.  Power  was  to  be  transmitted  to  a  distance  of 
approximately  30  miles  from  the  central  station.  There  were 
three  routes  available  for  the  new  line  no  one  of  which  had  any 
evident  marked  advantage  over  the  others.  Each  one  included 
a  different  amount  of  private  right-of-way,  of  old  pole  line,  of  new 
construction,  of  corners,  and  of  distance  over  which  the  construc- 
tion cost  would  be  excessive  on  account  of  high  trees  and  other 
obstructions.  The  comparative  economy  of  the  three  routes  was 
displayed,  by  use  of  the  adjusted  annual  costs  explained  above, 
and  tabulated  as  follows: 

TABLE  5. — ADJUSTED  ANNUAL  COSTS 


Item 

Route  A 

Route  B 

Route  C 

Miles 

Miles 

Miles 

New  construction  at  $350 
per  mile  

21 

8 

6 
9 

29 
3 

$  7,360 
1,710 

210 
342 

3,190 

19 

12 

4 

7 

31 
2H 

$  6,650 
2,560 

140 
266 

3,410 

22 
6 

10 

7 

28 
3 

$  7,700 
1,280 

350 
266 

3,080 

Reconstruction  on  old  poles 
at  $213  per  mile  
Right-of-way    at    $35    per 
mile 

Corners  at  $38  each  
Energy  loss,  4,000  kw.  load 
at  $110  per  mile  
Difficult  construction 

Total 

$12,812 

$13,026 

•• 

$12,676 

If  is  evident  that  Route  C  is  the  most  economical  even  though 
it  requires  the.  purchase  of  more  private  right-of-way  and  uses 
less  of  the  old  pole  line,  since  it  is  somewhat  shorter  and  some 
corners  are  eliminated.  The  extra  cost  of  difficult  construction  is 


108  ECONOMICS  OF  ELECTRICAL  DISTRIBUTION 

rather  hard  to  arrive  at  without  a  detailed  layout  of  the  line. 
It  can  be  estimated  however.  In  the  above  case,  the  amount  of 
extra  construction  was  so  nearly  equal  in  each  case  that  this  was 
not  thought  necessary. 

As  has  been  explained  in  the  previous  chapter  there  are  some 
other  features  of  a  line  that  cannot  be  included  in  such  a  cost 
analysis  and  which  might  have  considerable  weight  in  the 
determination  of  a  route.  One  route  might  be  more  accessible 
for  patrolling  and  repairs  than  the  others.  The  proximity  to  a 
railroad  might  affect  the  construction  cost  considerably.  The 
shelter  afforded  the  line  against  severe  storms  might  be  a  large 
factor  for  consideration.  These  points  can  however  be  weighed 
against  the  advantage  in  cost  alone  and  the  most  advantageous 
route  will  usually  be  evident. 

In  the  foregoing  discussion,  there  has  been  treated  only  a  few 
of  the  many  problems  involving  reconstruction  which  are  en- 
countered by  the  engineer.  However  it  has  been  attempted  to 
set  forth  the  underlying  principles  of  the  analysis  of  costs  on  such 
problems  so  that  they  can  be  applied  to  any  similar  problems 
with  modification  to  suit  the  particular  case.  The  curves  and 
figures  given  herewith  are  not  intended  for  use  under  any  other 
conditions  than  those  for  which  they  were  derived.  They 
merely  serve  as  an  example  of  how  the  methods  used  may  be 
applied  to  a  given  case.  The  outstanding  feature,  which  is 
recognized  after  the  application  of  these  methods  to  a  few  specific 
problems,  especially  problems  of  relative  economy,  is  the  great 
advantage  ordinarily  gained  by  shortening  a  line.  As  a  rule  any 
ordinary  extra  construction  cost  is  justifiable  if  the  line  is  notice- 
ably shortened  thereby.  This,  therefore,  points  to  the  advantage 
of  greater  care  in  the  selection  of  the  routes  for  new  lines  in  new 
territory  even  though  at  the  time  it  may  seem  advisable,  on 
account  of  light  load,  to  keep  the  construction  cost  a  minimum. 


CHAPTER  XI 
POWER  CIRCUITS 

PROBLEMS  RELATING  TO  LINES  CARRYING  POWER  LOAD  CHIEFLY 
— VOLTAGE — ECONOMICAL  CONDUCTOR  SIZE — USE  OF  Two 
LINES  IN  PLACE  OF  ONE — DISTRIBUTION  OF  LOAD 
OVER  SEVERAL  LINES 

The  study  of  primary  lines  will  be  divided  into  two  parts,  i.e., 
the  consideration  of  circuits  carrying  power  chiefly  and  of  those 
devoted  largely  to  lighting  load.  It  is  realized  that  on  most 
systems,  there  is  no  sharp  division  between  these  two  classes. 
In  most  cases,  where  the  power  load  is  comparatively  small, 
power  and  lighting  are  both  carried  on  the  same  lines.  When 
larger  power  loads  come  on,  the  difficulties  of  regulation  usually 
call  for  a  separation  of  circuits,  even  though  the  load  factor  on 
the  lines  is  thereby  reduced  somewhat.  The  facts  that  lighting 
load  requires  a  closer  regulation  than  power,  and  that,  when 
large  power  loads  are  considered,  lighting  and  power  load  usually 
overlap  considerably  during  the  heavy  loading  season,  justify 
such  a  practice. 

On  some  systems,  the  power  and  lighting  circuits  are  kept 
entirely  distinct.  On  others,  where  three-  or  two-phase  circuits 
are  used  for  lighting,  small-  and  medium-sized  power  loads  are 
taken  on  the  same  circuits,  while  separate  lines  are  run  for  large 
power  loads.  In  the  first  case,  it  is  very  often  the  practice  to  run 
a  three-phase  line  to  some  central  feeding  point  and  there  sepa- 
rate the  phases  into  individual  single-phase  circuits  for  lighting. 
In  the  latter  case,  all  branches  carrying  lighting  only  are  ordina- 
rily single-phase.  It  appears,  therefore,  that  the  problems  relating 
to  power  only,  to  power  and  lighting  combined  and  to  lighting 
only  may  be  very  similar  up  to  a  certain  point,  differing  only  in 
load  factor.  On  the  other  hand,  if  lighting  only  is  considered, 
the  quite  definite  load  factor  simplifies  the  study  somewhat. 
Also  the  single-phase  lines  are  problems  in  themselves.  This 
chapter  will  be  devoted  to  the  problems  of  power  circuits. 

Kind  of  Problems  Encountered. — The  questions  arising  in 
connection  with  power  circuits  have  to  do  mostly  with  voltage 
and  conductor  size.  Such  lines  are  nearly  always  run  on  roads, 
streets,  alleys  or  lot  lines  and  the  pole  spacing  and  location  is 
limited  by  the  mechanical  strength  of  the  pole,  by  the  arrange- 

109 


110  ECONOMICS  OF  ELECTRICAL  DISTRIBUTION 

ment  of  street  and  lot  lines,  by  provisions  for  future  extensions, 
etc.  Pole  heights  are  governed  by  standard  practice,  by  city 
ordinances  and  by  heights  necessary  to  clear  obstructions. 
Occasionally  there  may  arise  questions  as  to  the  economy  of  using 
private  right-of-way  instead  of  the  public  highway  for  short 
distances  but  such  problems  are  usually  small  ones  and  are  easily 
solved  by  a  comparison  of  the  annual  cost  of  the  two  alternatives. 
Voltage. — The  voltage  for  use  on  power  circuits  is  usually  a 
development  from  past  practice,  although  it  is  often  found 
economical  to  increase  the  voltage  when  the  load  increases 
beyond  a  certain  amount.  The  voltages  in  common  use  have 
been  pretty  well  standardized  at  2,200,  4,400,  6,600  and  11,000 
volts.  There  is  a  tendency  at  present  to  go  even  higher  for 
power  lines  with  heavy  loads,  but  such  lines  partake  more  of  the 
nature  of  transmission  lines.  On  any  line,  the  higher  the  voltage 
the  less  the  line  losses  and  the  larger  the  loads  which  can  be 
carried  on  the  line  with  a  given  regulation.  On  the  other  hand, 
the  cost  of  insulation  of  the  line  and  the  cost  of  transformers  is 
increased.  The  additional  precautions  which  it  is  necessary 
for  construction  men  to  take  in  working  with  a  higher  voltage  is 
also  a  factor  to  be  considered.  To  give  a  comparison,  6, 600- volt 
transformers  can  be  obtained  for  about  18  or  20  per  cent  more 
than  2,200-volt.  Thereby  the  voltage  is  multiplied  by  three, 
hence  the  line  loss  is  divided  by  nine  for  any  given  load  and  wire 
size.  Or,  for  the  same  per  cent  voltage  drop  and  conductor,  nine 
times  the  load  can  be  carried.  Where  loads  are  not  heavy,  such 
an  increase  in  capacity  may  not  be  desirable  as  compared  with 
the  increased  cost  of  construction.  Where  heavy  loads  are 
handled,  however,  a  voltage  of  6,600  or  11,000  may  often  be 
found  very  advantageous.  When  the  problem  is  one  of  chang- 
ing the  voltage  of  a  system  already  in  operation  to  a  higher  volt- 
age, the  cost  of  making  the  change  must  be  taken  into  account. 
The  old  line  transformers  must  be  disposed  of,  or  the  change 
made  gradually,  using  the  old  transformers  in  certain  districts 
until  they  are  worn  out.  Station  transformers  and  other  appa- 
ratus, suitable  for  the  higher  voltage,  must  be  provided.  Often, 
cables  must  be  replaced  with  those  of  higher  rating.  The  increase 
in  annual  charges  due  to  all  these  items  must  be  carefully  studied 
in  connection  with  the  value  of  the  increased  capacity  and  reduc- 
tion in  losses  achieved,  in  order  to  determine  the  economy  of  any 
such  alteration.  In  this  connection,  the  probable  increase  of 


POWER  CIRCUITS  111 

load  on  the  system  for  some  time  in  the  future  must  be  esti- 
mated and  what  further  changes  will  have  to  be  made  eventually 
to  care  for  probable  future  conditions. 

Voltage  Drop. — The  problem  of  voltage  drop  is  an  important 
one  to  consider  in  connection  with  power  circuits.  The  allow- 
able regulation  at  the  customer  is  more  or  less  fixed  by  consid- 
erations of  good  service  or  by  contract.  The  substation  bus 
voltage  may  be  kept  within  certain  known  limits.  The  question 
is  then  one  of  whether  to  serve  the  customer  by  a  circuit  of  small 
or  medium-sized  conductor  with  a  regulator,  or  of  large-sized 
conductor  without  a  regulator.  The  annual  cost  of  the  instal- 
lation as  a  whole,  including  cost  of  energy  losses,  will  be  the 
criterion.  On  a  large  system  with  a  steadily  increasing  load  it  is 
often  the  practice  to  standardize  on  one  or  two  conductor  sizes. 
A  new  line  is  built  of  standard  size  and  allowed  to  operate  at  low 
loads  unregulated.  Load  is  added  from  time  to  time  until, 
when  it  becomes  too  heavy,  a  regulator  is  added.  From  a 
practical  standpoint  this  method  has  its  advantages.  A  study  of 
the  economy  of  the  installation,  however,  will  still  be  of  real 
advantage  in  indicating  the  standard  sizes  to  use,  when  the  lines 
become  loaded  beyond  the  economical  limit,  etc. 

Power-factor  Improvement. — The  question  of  power  factor  is 
a  prominent  one  at  present.  Aside  from  the  proposition  of 
inducing  the  customer  to  improve  his  power  factor  by  using  that 
as  a  basis  for  rates,  there  is  a  further  interesting  problem  for  the 
central  station.  Poor  power  factor  means  increased  losses  for 
the  same  delivered  load  in  kilowatts.  There  will  be,  then,  a 
point  at  which  it  will  be  economical  to  install  static  or  synchron- 
ous condensers  in  order  to  reduce  these  losses  by  improving  the 
power  factor.  This  point  can  be  determined  by  a  comparison  of 
the  annual  cost  of  the  condenser  in  place,  with  the  value  of  the 
energy  conserved,  considering  also  the  improvement  in  regulation. 

Most  Economical  Conductor  Size. — The  determination  of  the 
most  economical  size  of  wire  for  any  load,  is  closely  connected 
with  the  considerations  of  voltage  and  voltage  drop,  as  is  evident 
from  the  above  paragraphs.  If  the  most  economical  conductor 
size,  considering  the  line  only,  is  known,  however,  it  serves  as  a 
starting  point  for  the  further  study  of  the  economy  of  the  circuit 
as  a  whole,  including  regulators,  etc.  An  example  will  be  given 
here  of  the  method  of  attacking  this  problem  of  most  economical 
wire  size  for  power  circuits. 


112  ECONOMICS  OF  ELECTRICAL  DISTRIBUTION 

Annual  Cost  Equation.  —  For  a  three-phase  line  the  annual  cost 
per  1,000  ft.  of  line 

=  g  (cost  of  wire  +  cost  of  stringing) 
+  0  (cost  of  poles,  fixtures,  guys,  etc.) 
-f  cost  of  energy  loss. 

Where  g  =  per  cent  interest,  taxes,  depreciation,  etc. 
(Cost  of  right-of-way  is  omitted  as  it  is  not  always  present  on 
such  lines,  and,  in  any  case,  is  the  same  for  all  sizes  of  conductor.) 

In  this  case,  instead  of  developing  the  above  equation  in  terms 
of  the  cross-sectional  area  of  conductor  and  then,  by  means  of  the 
first  derivative,  determining  the  most  economical  size,  it  has  been 
found  more  useful  to  investigate  the  range  of  loads  for  which 
any  given  stock  wire  size  is  more  economical  than  any  other. 
The  method  used  is  as  follows: 

The  annual  cost  per  1,000  ft.  of  line  for  each  standard  conduc- 
tor size  is  obtained  in  terms  of  the  cost  of  copper,  the  load, 
voltage,  power  factor,  equivalent  hours,  and  cost  of  energy. 
This  equation  is  of  the  form 

C.  (39) 


Where  Y  =  annual  cost  per  1,000  ft.  of  line, 
Ccu  =  cost  per  pound  for  copper, 
kw  =  load  in  kilowatts, 

E  =  voltage, 
Cos  0  =  power  factor, 

t  =  equivalent  hours, 
Ce  =  cost  of  energy  per  kilowatt-hour, 
KI,  K2,  Ks  =  constants. 

Combined  Equation.  —  If  the  equation  for  any  stock  size  of  wire 
is  combined  with  that  of  the  next  adjacent  size  by  equating  the 
annual  costs,  another  equation  is  obtained  which  expresses  the 
conditions  under  which  there  is  no  choice  in  economy  between 
the  two.  If,  for  example,  in  the  above  equation  E,  and  Ccu  are 
fixed,  the  combined  expression  would  give,  for  any  value  of 
tCe,  the  load  at  which  the  economy  changes  from  one  size  of 
conductor  to  the  next.  If  such  expressions  are  determined  and 
plotted  for  No.  6  to  No.  4  and  for  No.  4  to  No.  2,  for  example, 
the  values  of  kw/cos  6  between  the  two  curves,  for  any  value  of 
tCe,  indicates  the  range  of  loads  for  which  No.  4  wire  is  more 


POWER  CIRCUITS 


113 


economical  then  either  adjacent  size.     For  smaller  loads,  No.  6 
is  more  economical,  for  larger  loads,  No.  2  (see  Fig.  27). 

Value  of  Constants. — In  the  above  equation  (39)  the  constants 
KI  and  K%  depend  on  wire  size,  local  costs  for  stringing  wire, 
and  local  standards  of  construction  and  costs  for  use  with  each 
wire  size.  It  may  be  assumed  in  this  case  that  the  poles  will 
be  the  same  for  any  size  of  conductor,  and  hence  their  cost  will 
cancel  out  when  the  equations  for  two  wire  sizes  are  combined. 


Drawn  far46OO  Volis. 

Maybe  used  for  ZZOOvoffs  if  had 
iven  is  mulfipl/ecf  by  2 
opper  af  3O  cerrhs  per  pound 
For  values  ofiCe  see  Table  H 


200  400  GOO  800  1000  1200  1400  1600  1800 


FIG.  27. — Economical  wire  size  for  three-phase  primary. 


The  cost  of  pole  fixtures  (crossarms,  etc)  does  not  increase  pro- 
portionally with  wire  size  but  will  be  the  same  for  several  sizes 
and  then  change  abruptly  for  the  next  larger  group.  The  cost 
of  conductor  in  place,  including  incidental  material  and  labor 
cost  of  stringing,  will  follow  such  an  expression  as  Ka  +  Kb  Ccu, 
Ka  and  Kb  being  separately  determined  for  each  size  of  wire. 
These  constants,  properly  combined  give  the  values  of  KI  and  K2. 
Cost  of  Energy  Loss.  —  The  cost  of  energy  loss  is  determined 
from  the  equation:  Annual  charge  for  energy  loss  per  1,000  ft.  = 

C  I     kw     \  2 

372r  X  365  X  t  X  =  365,000  rtCe  (40) 


Where  r  =  resistance  of  conductor  per  1,000  ft. 

8 


114  ECONOMICS  OF  ELECTRICAL  DISTRIBUTION 

Then  K3  (in  Eq.  39)  =  365,000  r,  for  any  size  of  conductor. 

Numerical  Example. — The  following  equations  of  total  annual 
cost  were  obtained  in  a  specific  instance,  with  all  constants 
evaluated. 


TABLE  6.  —  EQUATIONS  OF  TOTAL  ANNUAL  COST 

SIZE  OF 
WIRE 

(Icin      \  2 
*pg   J      tCe 

4  8.41+    80.0Ccu+    92,500(  J™     Y  tCe 

\Hi  COS  u/ 

2  11.30  +  126.0CCU  +    58,100  %Ce 


0  12  .  43  +  204  .  OCCM  +    37  ,  300  (  -,kw    \  *tCe 

\Mi  COS  v/ 

00  14  .  62  +  251  .  OCcu  +    29  ,500  (-,fcM?  J)  \Ce 

\Ei  COS  P/ 

000  16  .  43  +  313  .  OCcu  +  23  ,  400  (  -,kw  }  *  tCe 

\Jti  cos  6/ 

0000  17  .  32  +  382  .  OCCM  +    18  ,  600  (     *w    \  *  tCe 


Equating  the  expressions  for  cost  for  each  adjacent  pair  of 
wire  sizes,  the  following  expressions  are  obtained. 

TABLE  7 

SIZE  OF  SIZE  OF 

WIRE  WIRE 

6        to         4  54,60o(Ffc^   Y  tCe  =    23.8  Ccu  +  1.71 

\.Cf  COS  (// 

4        to         2  34,400^^    \  2  tCe  =    46.1CCM  +  2.91 

2        to         0  20,800 (-,kw   .) 2  tCe  =    78.0C™  +  1.13 

\is  cos  oi 

0        to       00  7,8( 


00        to      000  6>100Vgcosg/    <Ce  =    61.2CCM  +  1.81 

000        to   0000  4,800^^g    \  *  tCe  =    69.0CCM  +  0.89 

0        to   0000  18 , 700  (     kw    )  *  tCe  =  178 .  OCCU  +  4 . 89 

\/i    COS    U/ 

Assuming  the  voltage  of  4,600  and  two  prices  for  copper,  30  cts. 
and  20  cts.,  which  represent  a  good  range  of  values,  the  equations 
become 


POWER  CIRCUITS  115 

TABLE  8.—  FOR  4,600  VOLTS 


SIZE  OF  SIZE  OF 

WIRE  WIRE 

6  to  4 

4  to  2 

2  to  0 

0  to  00 

00  to  000 

000  to  0000 

0  to  0000 


30-c 

tCe  = 
iC1  - 

T.  COPPER         30-CT.  COPPER 
3,430                2,510 

(kw/cos  0)2 
10,290 

(kw/cos  0)2 
7,460 

tC1 

(kw/cos  0)2 
24,900 

(kw/cos  0)2 
17,100 

tCe  = 
tCe  = 

fC 

(kw/cos  0)2 
41,800 

(kw/cos  0)2 
28,900 

(kw/cos  0)2 
70,000 

(kw/cos  0)2 
48,750 

(kw/cos  0)2 
95,200 

(kw/cos  0)2 
64,750 

tCe  = 

(kw/cos  0)2 
66,000 

(kw/cos  0)2 
45,800 

(kw/cos  0)2 

(kw/cos  0)2 

For  2,300  volts  the  numerators  of  the  above  expressions  for 
4,600  volts  should  be  divided  by  4 


(as  No.  6  to  No.  4.  tt7.  - 


The  curves  plotted  for  4,600  volts  can  be  used  for  2,300  volts  if 
the  given  load  for  2,300  volts  is  multiplied  by  2  before  applying 
curve. 

Plotting  Results  in  Curves.  —  As  was  mentioned  before  in 
Chap.  IX  on  "  Secondary  Transmission  Lines,"  the  cost  of  energy 
losses  per  kilowatt-hour,  Cej  and  the  equivalent  hours,  t,  are  inter- 
related. For  any  type  of  load,  such  as  that  on  a  typical  power 
circuit,  the  value  of  Ce  corresponding  to  any  value  of  t,  may  be 
determined  approximately.  For  power  circuits  in  heavily 
loaded  districts,  such  as  the  industrial  areas  in  a  large  city,  the 
load  factor,  equivalent  hours,  and  power  factor  may  be  practically 
the  same  for  nearly  all  lines.  In  that  case,  the  problem  is  simpli- 
fied and  the  range  of  loads  for  which  any  wire  size  is  most  eco- 
nomical is  more  simply  defined.  For  the  general  case  of  power 
circuits,  however,  the  load  may  vary  from  a  single  motor  to  a 
large  manufacturing  plant  load,  from  a  few  hours  per  week  oper- 
ation to  continuous  24  hr.  per  day.  Naturally  the  load  factor 
will  vary  through  a  large  range,  and  the  equivalent  hours  for 
the  same  load  factor  will  be  different  for  different  loads,  depending 
on  the  operation. 

The  above  equations  therefore  have  been  plotted  using  tCe 
as  one  coordinate,  Figs.  27  and  28.  The  method  of  determining 


116 


ECONOMICS  OF  ELECTRICAL  DISTRIBUTION 


the  value  of  tCe  for  any  load  will  be  explained  later.  In  order  to 
make  the  curves  applicable  to  loads  of  various  power  factors,  the 

other    coordinate    was    made  --  -.    The    computation   of  —  - 

cos  6  cos  6 

for  any  value  of  either  quantity  can  be  made  graphically  by  use 
of  the  curves  on  the  lower  half  of  the  figure.  The  intersection  of 
the  curve  for  any  load  with  the  horizontal  for  the  desired  power 


factor  gives  the  value  of 


COS  u 


on  the  scale  below. 


Drawn  for4600Vo!rs. 
May  be  used  for  23  OO  vo  Its  if  load 
giver  is  multiplied  bu  2 
—  Copper  afzo  cents  per  pound 
For  values  offCesee  fable  H 


1600  1800 


FIG.  28. — Economical  wire  size  for  three-phase  primary. 

Use  of  Curves. — Since  the  equations  were  so  developed  as  to 
show  the  points  where  economy  changes  from  one  size  of  wire  to 
another,  it  follows  that  the  area  between  two  curves  is  the  locus 
of  all  points  for  which  the  size  of  wire  shown  is  most  economical. 

Jci/j 

Thus,  on  Fig.  28,  for  (^  =  600  and  tCe  =  0.06,  No.  0  primary 

is  more  economical  than  No.  2  or  No.  00.  Similarly  it  is  more 
economical  than  No.  2  for  all  values  of  tCe  greater  than  .045  and 
is  more  economical  than  No.  00  for  all  values  of  tCe  less  than  .08. 
The  dotted  curve  shows  the  division  between  No.  0  and  No.  0000 
which  can  be  used  in  case  No.  00  and  No.  000  are  not  used  as 
standards  for  such  lines.  Two  sets  of  curves  are  given,  one  for 


POWER  CIRCUITS 


117 


20  ct.  copper  and  one  for  30-ct.     Intermediate  values  can  be 
interpolated. 

The  use  of  the  curves,  then,  is  as  follows : 

1.  Locate  the  intersection  of  the  curve  for  the  load  in  kilowatts  with  the 
horizontal  of  its  power  factor. 

2.  Locate  the  intersection  of  the  ordinate  through  this  point  .with  the 
proper  value  of  tCe  (scale  on  left). 

3.  The  area  in  which  this  point  lies  indicates  the  most  economical  wire 
size.     The  distance  of  the  point  from  the  curve  dividing  that  area  from  the 
next  adjacent  area  is  an  indication  of  the  amount  of  economical  advantage 
of  the  one  size  over  the  other. 

Determination  of  tCe.— The  proper  value  of  tCe  to  use  for  any 
load  may  be  determined  as  follows:  It  was  shown  in  Chap.  V 
in  the  discussion  of  equivalent  hours,  that  the  value  of  equivalent 
hours  corresponding  to  any  load  factor  may  vary  between  certain 
limits.  The  upper  limit  (load  factor  X  24)  would  be  correct 
only  for  a  load,  such  as  a  single  motor,  which  has  a  constant  value 
for  its  whole  time  of  operation.  The  minimum  value  would  be 
(load  factor)2  X  24  for  a  load  with  a  momentary  peak  and  the 
remainder  of  the  day's  curve  flat.  With  power  loads,  it  is  prob- 
able that  t  varies  from  somewhere  near  the  first  quantity  for 
small  loads,  such  as  one  or  two  motors,  to  somewhere  near  the 
average  between  the  two  for  large  loads  with  a  number  of  motors 
not  running  simultaneously.  Probably,  for  most  loads  encoun- 
tered, t  will  be  nearer  the  latter  figure. 

The  limits  of  t  would  be  as  follows: 

TABLE  9 


Load  factor 

Continuous  load 

Momentary  peak 

Average 

0 

0 

0 

0 

.10 

2.4 

.24 

1.32 

.20 

4.8 

.96 

2.88 

.30 

7.2 

2.16 

4.68 

.40 

9.6 

3.84 

6.72 

.50 

12.0 

6.0 

9.00 

.60 

14.4 

8.64 

11.52 

.70 

16.8 

11.76 

14.28 

.80 

19.2 

15.36 

17.28 

.90 

21.6 

19.44 

20.52 

1.00 

24.0 

24.00 

24.00 

118 


ECONOMICS  OF  ELECTRICAL  DISTRIBUTION 


An  approximate  determination  of  the  variation  of  the  cost  of 
energy  losses  with  load  factor  can  be  made  quite  easily  as  indi- 
cated in  "Appendix  A."  Of  course,  if  more  accurate  cost  figures 
have  been  determined  they  are  preferable.  The  following  indi- 
cates such  a  characteristic  variation. 


TABLE  10 


LOAD  FACTOK 

.10 
.20 
.30 
.40 
.50 
.60 
.70 
.80 
.90 
1.00 


COST  OF  ENERGY  Loss 
PER  KILOWATT-HOUR 

.0278 
.0184 
.0150 
.0133 
.0121 
.0113 
.0107 
.0101 
.0096 
.0092 


0.24 
0.22 
0.20 
0.18 
0.16 
0.14 

o.io 

0.08 
006 
0.04 
0.02 
0 

Average  Hours  per  Day  (Including  Sundays) 
0           3           6           9           12           15           18         21         24 

CURVE  A  -For  loads  which  are  constarfr-during  time  ' 
of  operation  as  one  motor  with  steady  hadftimiting  case) 

CURVE  B  -  For  loads  which  have  a  short  peak  with  the 
remainder  of  curve  flat.  (Limiting  case}            ' 

<7 

CUR 

VEC-  Average  of  A  and  i 
of  average  loads 

3.  Fits 

\ 

most 

cases 

£ 

(/t 

\ 

)/ 

/    \ 

/ 

/ 

« 

V 

:/ 

ty 

' 

dSSi 

<&' 

\  / 

^4 

f 

7\ 
-M* 

^rai 

v 

W] 

/ 

i 

/ 

/ 

I 

\ 

/ 

^ 

7 

i 

'y 

\ 

! 

/ 

£ 

x  ! 

I 

i 

(~* 

^ 

; 

! 

i 

0      0.10     0.20     O.JO     0.40     0.50     0.60    0.70     0.80     0.90     1.00 

Load  Factor 
FIG.  29. — Curves  showing  values  of  tCz  for  power  loads. 

Using  these  figures  in  connection  with  the  table  given  above 
the  curves  shown  on  Fig.  29  are  plotted  which  show  the  values 


POWER  CIRCUITS  119 

of  tCe  corresponding  to  any  load  factor.  A  second  scale  is  shown 
at  the  bottom  giving  the  average  number  of  hours  per  day  of 
peak  operation,  corresponding  to  any  load  factor,  which  is  useful, 
especially  in  connection  with  small  loads.  If  the  load  factor  of 
a  load  is  known  and  the  approximate  shape  of  its  typical  curve, 
the  value  of  tCe  may  be  selected.  For  a  single  motor  averaging 
3  hr.  a  day,  for  example,  tCe  would  be  about  .072.  For  a  larger 
load  with  a  definite  peak  and  an  average  load  curve,  with  a  load 
factor  of  30  per  cent,  the  value  of  tCe  would  be  somewhere  near 
.08.  In  general  it  may  be  said  that: 

TABLE  11 


Small  power 

.LOAD 

FACTOR 

Oto  20 

LOAD  CURVES 
Continuous  to  average  .  . 

tCe 

.  0  .  04  to  0  .  06 

Medium  power  

10  to  30 

Near  average  

.  0.04  to  0.07 

Large  power  

20  to  40 

Near  average  

.  0  .  05  to  0  .  09 

These  figures  are  indicated  on  the  curves,  Figs.  27,  28,  and  by 
the  brackets  on  the  left.  For  lines  carrying  lighting  only,  the 
value  of  tCe  would  be  between  .03  and  .05,  being  nearer  the  former 
for  residence  lighting  only,  and  approaching  the  latter  figure  for 
heavy  store  lighting,  etc.,  with  a  fair  average  of  about  .04. 

For  a  load  which  combines  power  and  lighting  the  value  of 
tCe  will  depend  on  the  proportion  of  each.  Where  the  power 
predominates,  the  lighting  load  will  have  the  effect  of  increasing 
the  load  factor.  Where  lighting  predominates  the  power  load 
will  have  the  same  effect.  In  either  case  a  higher  value  of  tCe 
should  be  used  than  would  be  assumed  for  the  predominating 
type  of  load  alone.  The  amount  of  increase  must  be  estimated 
from  a  consideration  of  the  probable  load  factor  for  the  particular 
case. 

Similar  Curves  Should  be  Derived  Locally. — Curves  such  as 
those  illustrated  here,  may  be  worked  out  for  any  system,  using 
local  cost  figures.  (The  examples  given  here  must  not  be 
considered  applicable  to  any  but  the  system  for  which  they  were 
derived.)  By  their  use  the  most  economical  wire  size  for  any 
load  may  be  determined  and,  from  that  point  on,  the  problem 
becomes  one  of  obtaining  proper  regulation  in  the  most  econom- 
ical way. 

Power  Circuits  with  t  Constant. — As  was  indicated  above,  for 
power  circuits  in  heavily  loaded  manufacturing  districts  where 


120  ECONOMICS  OF  ELECTRICAL  DISTRIBUTION 

there  is  considerable  diversity  of  load  on  each  line,  but  the  general 
characteristics  of  all  loads  are  somewhat  similar,  the  value  of 
equivalent  hours  will  be  nearly  the  same  for  all  lines.  If  an 
average  value  of  t  is  determined,  the  problem  of  economical 
conductor  size  may  be  simplified  and  it  is  possible  to  study  the 
effect  of  variations  in  the  cost  of  copper  and  of  energy  to  better 
advantage. 

Determination  of  t. — The  determination  of  average  equivalent 
hours  for  a  number  of  circuits  was  explained  in  Chap.  V  with  an 
example  of  lighting  circuits.  For  power  circuits  the  method  is 
similar.  In  a  specific  instance,  bi-monthly  curves  were  taken 
for  a  number  of  power  circuits  for  13  months  and  the  equivalent 
hours  for  each  such  curve  in  terms  of  the  curve's  peak  was 
determined.  This  was  assumed  to  be  the  average  for  the  half 
month  covered.  Each  of  these  figures  was  reduced  to  a  value  of 
equivalent  hours  in  terms  of  the  year's  peak  by  multiplying  it  by 
the  square  of  the  ratio  between  the  peak  for  the  day  for  which 
the  figure  was  derived  and  the  year's  peak.  The  26  values  thus 
obtained  were  averaged  and  the  result  was  assumed  to  be  a  fairly 
accurate  figure  for  the  equivalent  hours  for  the  whole  year, 
considering  days  of  operation  only,  exclusive  of  Sundays  and 
holidays.  In  the  example  taken,  the  value  of  equivalent  hours 
was  found  to  be  9.69.  This  is  high  for  ordinary  purposes,  being 
obtained  at  a  time  of  high  production  during  the  war  period,  but 
will  serve  as  an  illustration  of  the  method. 

Equations  for  Annual  Cost. — The  formulas  for  annual  cost  with 
any  size  of  conductor  were  altered  somewhat  for  convenience, 
and  to  bring  out  another  method  of  representing  the  results. 
It  was  assumed,  as  an  approximation,  that  all  charges  on  con- 
struction, which  are  not  proportional  to  the  cost  of  copper,  are  the 
same  for  all  sizes,  and  hence,  cancel  out  when  the  equations  for 
two  sizes  are  combined.  The  error  thus  introduced  is  small  for 
any  two  sizes  near  together,  such  as  No.  6  and  No.  4,  but  becomes 
greater  for  such  combinations  as  No.  0  and  No.  0000.  In  any 
case,  however,  it  can  be  kept  in  mind  and  compensated  for  when 
using  the  resulting  curves. 

Then,  the  annual  charge  per  1,000  ft.  on  the  above  basis 

Y  =  ?L  (3,OOOwiCCtt)  +  I2  X  -f  X  3,000  X  t  X  300  X 


+  900/2  -£•  tCe  (41) 


POWER  CIRCUITS 


121 


for  a  conductor  weighting  Wi  pounds  per  foot  and  of  cross- 
sectional  area  A  i,  using  300  working  days  per  year  to  correspond 
to  t  as  derived  above. 

If  this  expression  is  combined  with  a  similar  one  for  a  conductor 
of  weight  Wz  and  area  A2)  the  resulting  equation  may  be  reduced 
to 

^cu    ==   ~t      7     T~         ~~~\  1    t  (42) 


If  p,  t,  and  g  are  considered  constant  and  known  for  any  pair 
of  conductor  sizes,  this  equation  may  be  reduced  to 

r 

cu    _      If    '[  2 

c7  : 

Curves  for  Economical  Wire   Size. — The   curves   shown  in 
Fig.  30  have  been  plotted  from  this  equation.     Their  use  is 


(43) 


Sec 


IU 

4?  60 


•6  20 


FIG.  30. — Economical  wire  size  for  any  ratio  of  copper  cost  to  energy  cost  for 
3<£  power  circuits,  equivalent  hours  =  9.69. 

similar  to  that  of  Figs.  27  and  28,  the  areas  between  the  curves 
belonging  to  the  conductor  sizes  indicated.  It  is  clearly  evident 
that,  as  the  ratio  between  copper  price  and  energy  cost  increases, 
by  copper  price  increasing  or  energy  cost  decreasing,  the  smaller 
wire  becomes  more  economical.  As  was  explained  above,  in 
using  the  curve  between  No.  0  and  No.  0000,  the  excess  cost  of 
stringing  the  larger  size  must  be  kept  in  mind.  This  would 
have  the  same  effect  as  an  increase  in  price  of  copper,  i.e.j  to  raise 
the  curve  somewhat. 

Use  of  Two  Lines  in  Place  of  One. — The  study  of  economical 
conductor  size  may  be  further  extended,  in  a  similar  manner,  to 


122 


ECONOMICS  OF  ELECTRICAL  DISTRIBUTION 


the  consideration  of  the  economy  of  the  use  of  two  lines  in  place 
of  one.  Assuming  that,  for  mechanical  reasons,  No.  0000  wire 
is  the  largest  size  which  is  used  as  a  standard  on  a  given  overhead 
system,  there  is  a  load  at  which  it  becomes  economical  to  use 
more  than  one  No.  0000  circuit.  Another  circuit  of  any  size  of 
conductor  might  be  added,  according  to  the  load.  For  simplicity, 
the  problem  here  will  be  limited  to  that  of  discovering  at  what 
load  two  No.  0000  lines  are  more  economical  than  one,  assuming 
the  same  conditions  of  load  as  in  the  previous  example,  with 
t  constant. 

JJ_A    $ 


-* 

i                                           l 

< 

3<f>  Power 
Line 

3 

8 

3<£  'Power 
Line 

—  > 

i                                   i 

•«- 

s,*«*~0     § 

5       &-S4  Primary 

fl 


FIG.  31. — Arrangement  of  wires  showing  average  line  conditions. 

In  this  case,  the  charges  on  construction  cannot  be  assumed  to 
be  the  same  for  both  conditions  since  twice  as  much  pole  space  is 
occupied  by  two  circuits  as  by  one.  In  some  cases,  this  would 
simply  cause  the  addition  of  an  extra  crossarm.  It  very  often 
occurs,  however,  that,  in  districts  where  such  large  loads  are 
found,  the  poles  are  heavily  loaded  and  all  available  pin  positions 
are  valuable.  In  such  a  case,  the  circuit  should  be  charged  with 
that  proportion  of  the  total  cost  of  poles  and  fixtures  equal  to 
the  proportion  of  the  total  available  pole  space  which  it  occupies. 
If,  for  example,  such  an  arrangement  as  shown  on  Fig.  31  is 
assumed  to  represent  average  line  conditions,  one  of  the  three- 
phase  lines  occupies  one-fourth  the  pole  space.  If  the  average 
cost  of  a  pole  and  four  crossarms  with  other  fittings  is  $60  the 
cost  of  pole  space  for  the  line  in  question  is  $15  per  pole,  or 
about  $150  per  1,000  ft. 


POWER  CIRCUITS 


123 


If  Cp  =  cost  of  pole  space  per  1,000  ft. 

and  Cc  =  cost  of  stringing,  conductor,  insulators  and  pins  per 
1,000  ft. 
The  total  annual  cost  for  one  circuit 

Fi  =  j-^(3,OOOu;Cctt  +  Cp  +  Cc)  +  900  72  jj-  tCe        (44) 
and  for  two  circuits 

+  Cp  +  Cc)  +  2  X  900  £} 2  ^  *Ce  (45) 


The  investment  cost  is  doubled  while  the  energy  loss  is  halved. 
Equating  YI  and  Y* 


C 


c)  =  45072 


(46) 


In  order  to  put  the  curves  in  the  same  form  as  those  for  single 

C 
lines,  this  equation  must  be  reduced  to  an  evaluation  for  -^r* 

Tf  I-  =  1  -I-      p 

*  3,OOOwCCtt 

fc)  =  45072  ^  tCe 


Ce       gwkA 


72 


(47) 


of 
GJ 


rwo#oooo 


100     120     140     160      180    ZOO   2tO    Z40    260 
Maximum  Load  in  Amperes 


300    3aO    WO    360 


FIG.  32. — Economical  circuits  for  any  ratio  of  copper  cost  to  energy  cost  for 

three-phase  power. 

If  p,  t,  g,  w  and  A  are  constant  and  k  is  evaluated  for  several 
values  of  CCU}  this  equation  can  be  plotted  in  a  series  of  curves  as 


124 


ECONOMICS  OF  ELECTRICAL  DISTRIBUTION 


shown  on  Fig.  32,  one  curve  for  each  value  of  Ccu.  This  could 
have  been  added  to  Fig.  30,  if  desired,  as  it  is  of  the  same  form. 

Improvement  of  Regulation  by  Two  Circuits. — An  interesting 
point  arises  in  connection  with  the  use  of  two  circuits  instead 
of  one  when  a  low  power  factor  is  encountered.  While  the  resist- 
ance drop  of  a  conductor  decreases  proportionally  with  the  in- 
crease in  size,  the  inductance  drop  decreases  slowly.  Hence,  for 
power  factors  below  a  certain  value,  little  improvement  in  regu- 
lation is  accomplished  by  increasing  the  size  of  conductor.  On 
the  other  hand,  the  addition  of  a  second  circuit  materially 
reduces  the  inductance  drop  and  hence  the  regulation. 

As  a  concrete  example,  compare  the  voltage  drop  on  a  No.  0000 
circuit  carrying  3,000  kw.,  7,500  ft.,  with  two  No.  0  circuits,  both 
at  4,600  volts,  three-phase,  with  28-in.  spacing.  The  resistance 
of  No.  0  wire  being  about  twice  that  of  No.  0000,  the  equivalent 
resistance  of  both  installations  is  about  the  same.  The  power  loss 
and  voltage  drop,  at  different  power  factors,  as  figured  by  the 
charts  in  Chap.  VII,  are  as  follows: 


TABLE  12 


Power  factor 

Power  loss 

Voltage  drop 

One  No.  0000 

Two  No.  0 

One  No.  0000 

Two  No.  0 

Per  cent 

Per  cent 

Per  cent 

50 
85 
95 

14.6 
5.03 
4.05 

14.25 
4.87 
3.97 

18.05 
8.78 
6.64 

10.97 
6.04 
5.05 

It  is  very  evident,  from  the  above,  that,  at  a  low  power  factor, 
considerable  advantage  in  regulation  is  gained  by  using  two 
circuits  of  small  conductor  rather  than  one  circuit  of  twice  the 
size.  This  would  be  economical  if  the  improvement  in  regulation 
is  worth  more  than  the  increased  cost  of  construction.  At  high 
power  factors,  the  advantage  disappears  since  the  voltage  drop  is 
practically  equal  to  the  resistance  drop. 

Economical  Distribution  of  Load  over  Several  Lines. — One 
more  typical  problem  will  be  included  in  this  discussion  of  power 
circuits.  In  cases  where  there  are  several  different  lines,  of 
different  lengths  and  conductor  size,  feeding  a  large  load,  it  is 


POWER  CIRCUITS  125 

desirable  to  determine  the  most  economical  division  of  the  load 
among  those  lines.     For  simplicity  consider  two  circuits  only. 
If     7  =  the  total  load  current, 

Ia  =  the  economical  current  on  line  a, 
Ib  =  the  economical  current  on  line  6, 
7  =  I a  -f  h  (approximately). 

Since  the  lines  are  already  in  place,  the  annual  charges  on 
construction  will  be  constant  for  each 

Ka  =  annual  charges  on  construction  for  line  a. 
Kb  =  annual  charges  on  construction  for  line  b. 
The  annual  cost  of  energy  loss  on  each  circuit  will  vary  with 
72  and  Ce,  and  with  the  resistance,  t  being  fixed. 
KcIa2CeRa  =  cost  of  energy  loss  in  line  a. 
KcIb2CeRb  =  cost  of  energy  loss  in  line  b. 
Then  the  total  annual  charges  on  line  a, 

FTT      _L    ~V    T   ir<    E>  fAQ\ 

a    =    J^a   +  Ac/a^Ce/Ia  (48) 

and  on  line  6, 

7jf        |       Jf    T   2/^    E>  //in\ 

6    =    J^-b   T~  J^cib^eKb  V*yJ 

and  on  the  total  installation, 

7-V     _L_   V  TT     _I_7T     _l_JT/^/'72E>     -LT^E^ 

—    J-  a     \      J-  b    —    J^a   "t"  -L*-l   ~T  J^c^e(J-a  tia   T  IbKb) 

Substituting  for  7b  its  value  7  —  70, 

Y    =    Ka   +  Kb   +  KcCe(la2Ra   +  #&(72    ~   27J   +  702)  (50) 

and  the  annual  cost  per  ampere  carried 


-2I»  +  TJ)     <51> 

The  condition  of  maximum  economy  is  reached  when  y-  be- 
comes a  minimum.     The  value  of  Ia  to  accomplish  this  may 

Y 

be  determined  by  taking  the  first  derivative  of  -=-  with  respect  to 

7a,  since  7  is  constant 

dY/I  /2Ra       _  ,2R_bT\ 

dla  6M  7  la  "  7      V  : 

7    -        Rb     I  (vn 

*a    —    r>     _i_  r>    ^  \O*) 

Ra  -\-tib 

Similarly 

to  =  p^ETff  '  (») 

/la   T  ^6 

and 

laRa    ~    IbRb  (54) 


126 


ECONOMICS  OF  ELECTRICAL  DISTRIBUTION 


In  case  more  than  two  lines  are  considered,  two  or  more  lines 
can  be  represented  together  as  one  equivalent  circuit  and  the 
same  mathematics  apply. 

From  the  above,  therefore,  the  rule  can  be  formulated  that: 

The  most  economical  distribution  of  a  load  over  several  cir- 
cuits is  effected  by  making  the  load  on  each  circuit  inversely 
proportional  to  its  resistance,  i.e.,  by  making  the  IR  drop  equal 
on  all  lines. 

This  is  what  might  have  been  expected  if  it  is  considered  that 
the  total  annual  charges  on  construction  will  be  the  same,  regard- 
less of  the  division  of  the  load,  and  hence,  the  object  is  to  make 
the  total  energy  losses  a  minimum.  The  accompanying  Fig. 
33  indicates  how  the  total  cost  of  energy  loss  on  two  circuits 


f  Energy  Losses  in  dollars 

Tofat  L 
Lima 

octd  **  400 
-400  ft  of  4 
-.  400  ft  of* 
Loss  at  * 
lent  Hour 
er  Voltage 
Factorof 

1/fw. 

SOMCMcai 
50  MC  Meat 
O.OIperkn 
,=/0 
=  4600v. 
'oad*80% 

•>le+SOOOt 
le  +5000  f 
-hr 

t.of#000 
>-.OftO 

? 

\ 

, 

Line  b  • 
&»*!, 
tguiva 
fleceiv 
Power 

\ 

| 

^ 

^ 

<§ 

\ 

1 

^ 

^ 

I 

finnn 

V 

J 

[— 

^^ 

-"" 

06  0.8  1.0 

Ratio  of  Load  on  Two  Lines  in  Amp«res=  ^ 

FIG.  33. — Total  annual  cost  of  energy  losses  for  various  methods  of  dividing  a 
load  between  two  lines. 

with  different  divisions  of  load  can  be  displayed  graphically. 
The  point  of  minimum  cost  is  clearly  defined. 

Division  of  Load  Between  Lines  in  Parallel. — For  direct- 
current  circuits  the  load  would  naturally  divide  itself  in  prac- 
tically the  most  economical  proportions  if  the  lines  are  paralleled. 
For  alternating  current,  however,  the  inductive  reactance  has 
an  effect,  and  the  natural  division  of  current  would  be  somewhat 
different.  The  currents  would  divide  so  that  the  ratio  of  the 
current  in  any  circuit  to  the  load  current  is  the  same  as  the  ratio 
of  the  admittance  of  that  circuit  to  the  combined  equivalent 
admittances  of  all  the  circuits.  For  example  with  a  No.  0  and  a 
No.  0000  overhead  circuit,  of  the  same  length,  spacing,  etc. 

For  No.  0,  RQ  =  .539  ohm  per  mile 
No.  0000,  #0000  =  .269  ohm  per  mile 


POWER  CIRCUITS'  127 

The  most  economical  division  of  current  would  then  be 
0  9fiQ 

7»  "  0.589  +0.269f  = 

0 
Q  539  +  Q  269 

The  natural  division,  if  the  circuits  were  paralleled  would  be 


70ooo  =  l  =  °<667/  °r  %  the  total  current 


70  =  ^ —  —  7  =  0.4447  or  44.4  per  cent  of  the  load  current 

Y    total 

7oooo  =  ^r^30  7  =  0.5667  or  56.6  per  cent  of  the  load  current 

Y  total 

The  fact  that  the  arithmetical  sum  of  70  and  70ooo  is  not  equal 
to  7  is  due  to  the  fact  that  they  are  not  in  phase  with  each  other, 
nor  with  7. 

Large  Economies  Possible  on  Power  Circuits. — The  field  of 
study  of  the  economics  of  power  circuits  will  be  found  quite 
extensive  and  very  fruitful  of  real  results.  Such  circuits  in 
manufacturing  districts,  usually  handle  loads  many  times  as 
large  as  those  on  the  ordinary  lighting  circuits,  single  loads 
amounting  to  several  thousand  kilowatts  in  some  places.  The 
load  factor  is  also  comparatively  high  in  most  cases.  The  cost 
of  pole  space,  on  the  other  hand,  is  often  considerably  less  per 
circuit  on  account  of  the  fact  that  power  circuits  are  usually 
run  on  poles  which  would  be  set  for  lighting  circuits  in  any  case. 
It  is  evident,  therefore,  that  the  saving  of  a  few  per  cent  in  line 
loss  on  such  a  line  may  mean  the  saving  of  a  considerable  sum  of 
money  during  the  year.  Hence,  the  operation  under  the  most 
economical  conditions  possible  is  probably  productive  of  more  real 
saving  in  money  than  on  any  other  part  of  the  system.  A  study 
of  the  economics  of  any  type  of  installation  is  always  beneficial, 
but  in  the  case  of  power  circuits  it  is  most  imperative.  The 
problems  exemplified  in  this  chapter  indicate  the  type  of  question 
of  this  kind  which  will  arise  most  often  and  illustrate  methods 
which  have  been  found  useful  for  attacking  their  solution. 


CHAPTER  XII 
LIGHTING  CIRCUITS 

ECONOMICAL  STUDIES  ON  CIRCUITS  CARRYING  LIGHTING  ONLY — 

PREDICTION  OF  LOAD — CONDUCTOR  SIZE — INCREASING 

CAPACITY    OF    OVERLOADED    SYSTEMS 

The  preceding  chapter  indicated  that  in  general  there  were 
three  classes  of  primary  circuits,  those  carrying  lighting  load 
only,  those  carrying  power  loads  only  and  those  carrying  loads 
made  up  of  a  combination  of  the  two.  It  is  here  planned  to  deal 
with  circuits  carrying  lighting  load  only — particularly  in  reference 
to  residence  lighting. 

The  problems  encountered  in  general  can  be  divided  into  three 
classes.  In  the  first  class  are  those  pertaining  to  the  design  of  a 
new  system  to  handle  a  given  or  predicted  load,  such  as  would  be 
found  where  it  is  planned  to  build  a  distribution  system  in  a 
town  where  no  electric  service  has  been  furnished  before.  In 
this  case  we  meet  a  problem  somewhat  similar  in  characteristics 
to  the  one  encountered  under  " backbone"  transmission  lines. 
Here  our  purpose  is  to  design  the  most  economical  system  possible 
with  few  of  the  factors  of  design  previously  established.  The 
only  limiting  features  would  be  accepted  practice,  equipment 
obtainable,  and  the  general  knowledge  of  the  subject  as  recorded 
in  other  work  of  the  same  class.  The  problem  would  there- 
fore resolve  itself  into  one  of  compiling  costs  on  materials  and 
labor,  and  working  out,  as  pointed  out  previously,  comparative 
costs  for  several  different  alternative  designs  using  different 
voltages,  different  types  of  primaries,  such  as  single-phase, 
three-phase,  four-wire,  etc.  The  method  of  determining  an 
economical  conductor  size  will  be  discussed  later  on  in  this 
chapter.  It  is  a  matter  of  finding  the  lowest  annual  charges 
(investment  and  energy  losses)  within  the  limits  of  the  quality  of 
service  desired. 

The  second  class  of  problems  would  be  that  of  operating  and 
extending  an  existing  system  of  primary  lighting  lines  in  the 
most  economical  manner.  This  class  of  problems  would  be  the 
most  commonly  encountered  in  practice.  Here  certain  limita- 
tions are  found  prescribed  by  the  characteristics  of  the  system 
at  hand  which  would,  in  general,  prevent  any  considerable  change 
from  the  practice  laid  out  at  the  inception  of  the  project. 

128 


LIGHTING  CIRCUITS  129 

However,  it  is  advisable  to  be  prepared  always  to  contemplate 
the  possibility  of  a  radical  change  in  such  a  system.  This  brings 
in  the  third  class  of  problems.  They  call  for  a  study  of  the 
advisability  of  making  such  changes  as  raising  the  voltage, 
changing  from  S  0  to  3  0,  three- wire  or  four- wire,  etc. 

Predicting  Load  on  Residence  Lighting  Circuits. — The  problem 
of  predicting  load  on  circuits  does  not  involve  economics.  How- 
ever, a  correct  estimate  of  what  should  be  expected  in  the  next 
following  years  is  essential  as  a  basis  for  economical  design. 

If  the  demands  for  electric  service  which  are  made  each  year 
are  to  be  met  economically,  they  must  be  anticipated  far  enough 
in  advance  to  enable  provisions  for  serving  to  be  made  them  when 
it  is  most  economical  to  do  so.  About  the  only  way  to  make 
intelligent  estimates  of  future  requirements  is  to  couple  an  analysis 
of  past  rates  of  load  increase  with  a  far-sighted  judgment  which 
will  take  into  account  the  effect  on  future  conditions  of  the  past 
rate  of  growth. 

Predicting  the  load  on  circuits  carrying  a  load  consisting  of 
residence  and  store  lighting  is  usually  simplified  by  the  fact  that 
the  growth  is  relatively  constant  and  that  the  other  factors  that 
come  in  to  modify  the  estimates  can  be  analyzed  and  allowance 
can  be  made  for  them. 

In  the  following  it  will  be  shown  how  an  analysis  of  the  con- 
ditions causing  the  increase  has  been  attempted  for  a  number  of 
single-phase  circuits  fed  by  200,000-circ.  mil.  underground  cables 
and  No.  0  or  No.  0000  overhead  wires. 

The  data  available  for  such  a  study  consist  of  a  set  of  curves, 
one  for  each  circuit,  showing  the  monthly  maxima  over  a  period 
of  4  years.  These  curves  are  not  all  of  the  same  outline,  yet 
there  are  certain  characteristics  appertaining  to  all  which  could 
be  expressed  in  one  curve  to  be  considered  typical.  A  typical 
curve  would  serve  as  a  fairly  firm  foundation  on  which  to  base 
estimates  for  the  future. 

The  changes  in  circuit  loads,  shown  by  an  inspection  of  the 
monthly  maxima,  are  caused  by  a  number  of  easily  defined 
factors.  The  first  is  a  seasonal  condition  due  to  the  change  in 
the  length  of  days  and  the  result  of  cold  weather  keeping  people 
indoors;  the  second  is  an  increase  in  the  use  of  electricity  by  each 
customer,  caused  by  the  greater  appreciation  of  the  uses  of 
electrical  energy  in  the  home;  the  third  is  due  to  increase  of 
population  within  the  boundaries  of  each  circuit;  the  fourth  is  due 


130 


ECONOMICS  OF  ELECTRICAL  DISTRIBUTION 


to  the  embracing  of  new  territory  by  the  circuit  or  to  the  loss  of 
old  territory  caused  by  switching  of  load  on  account  of  overload 
or  other  operating  necessities. 

The  first  two  factors  are  constants  in  their  own  particular 
sense — that  is  to  say,  the  rate  of  increase  due  to  these  factors  is  a 
fairly  constant  value — and  the  third  is  partly  so,  for  it  may  be 
subdivided  into  two,  the  factor  of  normal  increase  in  population 
and  that  of  abnormal  increase.  Normal  increase  will  be  encount- 
ered in  practically  every  section  of  the  city,  abnormal  only  in 
sections  not  yet  closely  built  up  and  to  which  people  are  attracted 
by  real  estate  activity  or  the  circumstances  of  industrial  develop- 
ment. The  fourth  factor  is  a  result  of  the  preceding  three  and 
niust  be  considered  as  indefinite,  as  it  appears  both  for  and 
against  increases  of  load.  It  is  a  factor,  however,  for  which 
correction  may  be  made,  since  its  use  lies  within  the  control  of 
the  operating  company. 


Ordincrtes  of  Curve  I 
—  i  oo  to  o  —  r- 

,.i-r,T 

r>  —i  o>  «o  o  = 

Ordincrres  of  CurveTI 

^x 

^"*" 

***« 



>-., 

; 

X 

X 

«^^ 

^ 

•  

> 

xv 

~7 

/  t 

/ 

X 

^ 

N 

N 
N 

s 
s 

/ 

x 

"\ 

'A 

-»-->OzdiDsoe        >       *2       >•       &       *^~ 
b       C  .     Q       M       £                 <        o-        <        z        Li       ^       Q- 

Sozo<£3i<2:52<£ 

FIG.  34. — Typical  curves  to  show  variation  in  monthly  maxima. 

The  ordinates  of  curve  I,  Fig.  34  are  proportionate  to  the 
sums  of  the  maximum  monthly  loads  on  the  circuits,  averaged 
over  3  years.  It  was  considered  that  the  indeterminate  factor  of 
changes  in  territory  would  not  affect  the  curve,  inasmuch  as  they 
appeared  in  different  circuits  at  different  times  of  the  year  and 
their  effect  would  be  reduced  to  a  comparatively  small  value 
with  an  average  equal  effect  on  all  the  ordinates  of  the  curve,  in 
view  of  the  long  period  of  time  covered.  This  assumption  is 
borne  out  by  the  comparison  of  curve  I  with  curve  II.  This 
last  curve  covers  a  period  of  only  one  year — and  was  corrected 
for  changes  in  territory.  The  corrections  were  made  by  deduct- 
ing from  the  totals  of  each  month  the  amount  representing  the 
load  in  the  same  territory  appearing  in  two  circuits  in  the  same 


LIGHTING  CIRCUITS  131 

month.  This  condition  of  duplication  of  load  is  due  to  the 
fact  that  when  a  change  in  boundary  for  the  relief  of  any 
circuit  is  made,  the  load  on  the  section  cut  off  appears  in  the 
maxima  of  both  the  circuits  relieved  and  relieving.  It  would 
be  possible  naturally  to  take  a  well-built-up  district  including 
several  circuits  and  establish  a  curve  showing  the  growth,  per 
year,  of  that  nature  of  load,  leaving  the  new  circuits,  with  abnor- 
mal growth,  for  a  special  study.  This  refinement  is  hardly 
necessary,  as  the  percentage  of  growth  would  not  be  much 
diminished.  In  the  method  used  here  there  is  introduced  a  small 
factor  of  safety. 

In  order  to  make  the  curve  more  easily  applicable  it  has  been 
reduced  to  a  table  of  percentages.  Table  13,  in  which  the  load 
for  each  month  in  the  year  appears  as  a  percentage  of  every  other 
month  in  the  year.  It  is  possible  by  the  use  of  this  table  to  take 
the  load  on  any  circuit  for  any  given  month  and  predict  on  that 
circuit  the  load  for  any  future  date,  always  considering  that  the 
boundaries  of  the  circuit  remain  unchanged  and  that  there  are  no 
particular  conditions  in  that  circuit  which  will  cause  an  abnormal 
increase  or  decrease  in  the  load. 

It  is  not  the  intention  that  this  curve,  or  the  percentage 
table  developed  from  it,  shall  figure  as  an  absolute  method  for 
estimating  future  loads,  but  merely  as  a  basis  on  which  are  to  be 
imposed  the  particular  conditions  obtaining  for  each  territory 
under  consideration.  The  table  contains  no  allowances  for 
abnormal  conditions,  and  the  results  derived  from  it  may,  in 
some  cases,  have  to  be  considerably  modified  by  such  conditions 
as  changing  of  circuit  boundaries,  rapid  increase  in  rate  of  settle- 
ment, and  others.  However,  the  table  is  sufficiently  accurate 
to  serve  as  a  basis  in  estimates  on  the  necessity  for  future 
work,  unless  the  speed  of  growth  of  the  city  is  greatly  diminished 
or  increased  from  the  average  rate  maintained  for  the  years 
considered. 

The  table  gives  the  load  on  any  circuit  for  any  month  in  terms 
of  any  other  month,  and  by  correct  selection  of  factors  it  is 
possible  to  predict  with  reasonable  accuracy  the  future  maximum 
load  in  amperes  on  any  circuit  if  a  present  or  previous  reading  on 
that  circuit  is  given.  For  example,  the  November  load  on  a 
circuit  is  223  amp.,  and  it  is  desired  to  estimate  its  load  for 
January,  two  years  later.  Taking  the  November  load  as  1  in 
the  table,  the  load  for  the  following  September  is  .993  of  the 


132 


ECONOMICS  OF  ELECTRICAL  DISTRIBUTION 


November  load;  then  taking  this  September  load  as  1,  the  load 
for  the  following  January  is  1.23  of  this  calculated  September 
load.  In  this  case,  therefore,  the  January  maximum  load  of 
this  circuit  will  be  223  X  .993  X  1.23  =  272  amp.  This  table 
should  be  read  downward  for  predicting  future  load,  upward  if 
past  loads  are  to  be  determined. 

TABLE  13. — RATIO  OF  EACH  MONTH'S  LOAD  TO  THE  TOTAL  LOAD  FOR  THE 
REMAINING  MONTHS  OF  THE  YEAR 


Sept. 

Oct. 

Nov. 

Dec. 

Jan. 

Feb. 

Mar. 

Apr. 

May 

June 

July 

Aug. 

Sept. 

Sept.... 

1.000 

0.892 

0.825 

0.815 

0.812 

0.831 

0.836 

0.862 

0.911 

1.003 

1.098 

0.988 

0.831 

Oct  

1.113 

1.000 

0.913 

0.907 

0.904 

0.925 

0.931 

0.958 

1.013 

1.117 

1.223 

1.100 

0.925 

Nov.... 

1   210 

1.087 

1.000 

0.986 

0.983 

1.006 

1.012 

042 

1   102 

1   216 

1    330 

1   196 

1.006 

Dec.  .  .  . 

1.226 

1.101 

1.012 

1.000 

0.996 

1.018 

1.026 

.057 

1.117 

1.231 

1.348 

1.212 

1.018 

Jan  

1.230 

1.105 

1.015 

1.003 

1.000 

1.022 

1.029 

.061 

1.121 

1.236 

1.352 

1.216 

1.022 

Feb  

1  .  203 

1.081 

0.993 

0.980 

0.977 

1.000 

1.006 

.037 

1.096 

1.208 

1.332 

1.188 

1.000 

Mar.  .  .  . 

1.195 

1.073 

0.986 

0.974 

0.972 

0.993 

1.000 

.030 

1.088 

1.200 

1.312 

1.182 

0.993 

Apr  .... 

1.159 

1.041 

0.9.57 

0.945 

0.943 

0.964 

0.970 

.000 

1.056 

1.163 

1.273 

1.146 

0.964 

May  .  .  . 

1.096 

0.986 

0.906 

0.895 

0.892 

0.912 

0.918 

0.946 

1.000 

1.100 

1.204 

1.083 

0.912 

June.  .  . 

0.995 

0.894 

0.822 

0.812 

0.809 

0.827 

0.833 

0.858 

0.906 

1.000 

1.093 

0.984 

0.827 

July.... 

0.910 

0.818 

0.752 

0.742 

0.739 

0.756 

0.761 

0.784 

0.829 

0.914 

1.000 

0.900 

0.756 

Aug.  .  .  . 

1.010 

0.908 

0.835 

0.825 

0.822 

0.841 

0.846 

0.872 

0.922 

1.014 

1.110 

1.000 

0.841 

Sept.... 

1.203 

1.081 

0.993 

0.980 

0.977 

1.000 

1.006 

1.037 

1.096 

1.208 

1.322 

1.188 

1.000 

Having  established  a  table  for  predicting  loads  it  is  applied 
to  the  planning  of  the  necessary  new  equipment  required  to  take 
care  of  the  expected  loads.  It  will  serve  not  only  as  a  basis 
for  designing  new  overhead  or  underground  feeders,  but  also  for 
substation,  transmission  and  power  line  requirements.  It  is 
evident  that  proper  prediction  of  load  is  of  great  importance  in 
the  work  of  economics  of  distribution  as  it  affects  the  growth 
of  the  system  as  a  whole.  Proper  care  in  compilation  of  data  and 
considerable  study  of  local  conditions  will  be  well  repaid  in 
bringing  about  the  possible  economies  of  the  system. 

Economical  Wire  Size  for  Single -phase  Lighting  Primaries.— 
In  extending  an  existing  system  the  problem  of  the  proper  size 
for  lighting  primaries  will  present  itself.  Loads  to  be  handled  will 
be  estimated  as  shown  above.  A  method  of  obtaining  the  equiva- 
lent hours  for  those  loads  was  given  in  Chap.  V.  With  these  two 
factors  known  we  can  proceed,  as  an  example,  to  determine  the 
economical  wire  size  for  single-phase,  4,600-volt  primaries.  In 
the  preceding  chapter  on  power  circuits  a  complete  analysis  of  a 


LIGHTING  CIRCUITS  133 

similar  problem  was  given.  Here  it  is  proposed  briefly  to  give 
the  equations  and  the  constants  that  apply  particularly  to 
lighting  circuits. 

For  single-phase  lines  the  cost  of  conductor  will  be  two-thirds 
that  for  three-phase.  The  cost  for  pole  fixtures  will  be  more 
than  two-thirds  on  account  of  the  fact  that  the  crossarm  cost  is 
included. 

The  following  equations  were  obtained  in  a  specific  instance. 


TABLE  14. — EQUATIONS  OF  TOTAL  ANNUAL  COST 


SIZE  OF 
WIRE 


6  5.70  +  37.5CCM  +  294,000  (  -,kw     \  *  tCe 

\rj  COS  a/ 

4  6.17  +  53.3CW  +  185,000  (     kw    )  '  tCe 

\J1/  COS  (// 

2  8.10+  84.0C™  +116,200  (  „  kw    V  tCe 

\E  cos  6  / 

(Icill        \   2 
E      ™s    J       tCe 

00  10.58  +  167.4CCU  +    59,000 


000  11.80  +  208.  5CCtt  +    46,800  (  „  kw    }*lCe 

] 


E  cos  0 

„  kw 
\E  cos  d 


0000 


12  .  39  +  255  .  OCCU  +    37  ,  200  (      *™   \  *  tCe 


Equating  the  expressions  for  cost  for  each  adjacent  pair  of 
wire  sizes  the  following  equations  are  obtained. 

TABLE  15 

SIZE  OF         SIZE  OB1 
WIRE  WIRE 

6     to        4  109,000    (0  c^  0)  ^  =     15.8CCM +  0.47 


4     to        2  68,800   (      "^    j     tCe  =    30 . 7CC«  +  1 . 93 

2     to         0  41,600   (.,  kw  y  tCe  =    52.0CCW  +  0.76 

\Jbj  cos  u / 

0     to       00  15,600  (_,  kw  Y  tCe  =    31.4CCU  +  1.72 

\xi  COS  "/ 

/        Jem     \  2 

00     to     000  12,200   („         J    tCe=    41.1Ceu  +  1.22 

\A  cos  0/ 

000     to  0000  9,600   (_  kw  Y  tCe  =    46.5CCW  +  0.59 

\ii  COS  0 J 

0     to  0000  37,400   (^  kw    \  *  tCe  =  119.0  Ccu  +  3.53 

\E  cos  0/ 


134 


ECONOMICS  OF  ELECTRICAL  DISTRIBUTION 


Assuming  the  voltage  at  4,600  volts  and  copper  at  30  cts.  and 
20  cts.  per  pound,  the  equations  become 


TABLE  16.— FOR  4,600  VOLTS 


SIZE  OF 
WIRE 


SIZE  OF 
WIRE 


6 

to 

4 

4 

to 

2 

2 

to 

0 

0 

to 

00 

00 

to 

000 

000 

to 

0000 

0 

to 

0000 

tC1     - 

30-CT.  COPPER 
1,010 

20-cx.  COPPER 

705 

ll^e    — 

tCe    = 

tce  = 

fC 

(Aw/cos  0)2 
3,420 

(kw/cos  0)2 
2,480 

(kw/cos  0)2 
8,320 

(kw/cos  0)2 
5,670 

(kw/cos  0)2 

15,100 

(kw/cos  0)2 
10,850 

tCe    = 
tCe    = 

tc.  = 

(kw/cos  0)2 
23,500 

(kw/cos  0)2 
16,380 

(kw/cos  0)2 
32,100 

(kw/cos  0)2 
21,800 

(kw/cos  0)2 
22,200 

(kw/cos  0)2- 
15,470 

For  2,300  volts  the  same  formulas  can  be  used  if  the  load  in 
kw  is  multiplied  by  two. 

The  factor  tCe  still  remains  to  be  evaluated  in  order  to  plot 
the  curves.  In  the  case  of  lighting  circuits  the  load  factor  can 
be  taken  as  practically  a  constant  and  the  shape  of  the  load  curve 
can  be  also  assumed  to  remain  uniform.  Hence  a  simple  value 
for  tCe  can  be  used  instead  of  keeping  it  a  variable  in  the  equation, 
as  was  done  with  power  circuits.  This  factor,  as  used  in  Fig.  35, 
was  assumed  as  .04.  The  equivalent  hours  for  lighting  were 
determined  previously.  The  cost  of  energy  combined  with  this 
was  found  to  give  the  above  figure  as  an  average. 

The  curve  is  plotted  for  two  values  of  the  cost  of  copper 
(20  cts.  and  30  cts.).  The  area  in  which  the  load  ordinate  inter- 
sects the  curve  for  copper  cost  indicates  the  wire  size  to  be 
used.  The  economical  advantage  of  the  size  given  is  indicated 
by  the  distance  to  the  adjacent  areas.  This  method  of  exhibiting 
results  has  the  advantage  of  showing  graphically  the  limits 
between  which  one  can  work  and  will  show  readily  the  effect  of 
an  increase  or  decrease  of  load. 

Use  of  Regulators. — In  connection  with  the  determination  of 
the  most  economical  wire  size  for  a  lighting  circuit  it  is  sometimes 
necessary  to  include  a  consideration  of  the  cost  of  a  regulator. 
As  with  power  circuits,  it  may  be  desirable  to  determine  whether 


LIGHTING  CIRCUITS 


135 


it  is  more  economical  to  use  large  conductors  without  a  regulator 
than  smaller  conductors  with  a  regulator.  The  annual  charges 
on  investment  and  losses  for  the  regulator  must  be  included  with 
those  of  the  line  in  this  case.  With  lighting  circuits,  however,  it 
is  usually  more  necessary  to  have  good  regulation  than  with 
power  circuits  to  prevent  fluctuation  in  the  illumination.  Hence 
it  is  often  preferable  to  use  a  regulator  in  any  case,  regardless  of 
the  exact  economy,  in  order  to  keep  the  voltage  at  or  near  the 
center  of  the  load  as  nearly  constant  as  possible. 

Other  Problems  on  Lighting  Circuits. — With  lighting  circuits 


200 


1000 


1200 


400  600  800 

Load  in  KW. 
FIG.  35. — Economical  wire  size  for  single-phase  primaries. 


there  appear  also  some  of  the  same  problems  met  with  in  the 
preceding  chapters, -such  as  the  economical  load  which  can  be 
carried  on  a  line  already  in  place,  when  an  additional  circuit 
should  be  installed  or  the  wire  size  increased,  etc.  As  the  solution 
is  not  essentially  different  from  that  for  other  types  of  lines, 
these  questions  need  not  be  discussed  in  detail  here. 

Increasing  Capacity  of  an  Overloaded  System. — It  has  been 
the  case  in  many,  if  not  all,  large  cities,  and  also  in  a  number  of 
small  communities,  that  the  systems  of  lighting  circuits,  which 
a  few  years  ago  were  apparently  perfectly  satisfactory  for  a  long 
time  to  come,  have  become  inadequate  on  account  of  the  large 
increase  in  the  utilization  of  electricity  and  in  view  of  the  prospec- 


136  ECONOMICS  OF  ELECTRICAL  DISTRIBUTION 

tive  further  increases  in  the  future.  This  is  especially  true  where 
comparatively  low  primary  voltages  are  in  use,  such  as  1,100  or 
2,300  volts,  and  where  single-phase  circuits,  running  from  the 
usual  type  of  main  sub-stations,  is  the  practice.  The  change  to 
be  made  in  such  a  system  is  a  problem  warranting  the  most 
careful  consideration.  It  is  quite  probable  that  future  develop- 
ments may,  in  a  few  years,  make  any  provision,  which  could  now 
be  made  in  the  light  of  present  conditions,  again  inadequate. 
This  cannot  be  foreseen  as  yet  however.  The  best  that  can  be 
done  is  to  provide  for  an  increase  at  the  same  rate  as  at  present 
for  a  reasonable  number  of  years,  leaving,  if  post,  !e,  a  good 
opportunity  for  further  changes  at  the  end  of  that  time,  to  care 
for  the  unforeseen  developments.  Several  me  ods  of  providing 
for  this  increase  in  capacity  have  been  tried  in  different  cities 
with  good  success.  The  principal  ones  will  bt  noted  i  :re  with 
brief  notes  on  the  advantages  claimed  and  some  factors  which 
affect  the  making  of  such  changes. 

Probably  the  simplest  method  is  that  of  increasing  the  voltage 
used.  'The  advantage  of  such  a  change  was  mentioned  in  the 
chapter  on  "  Power  Circuits,"  i.e.,  the  increase  in  load  carried, 
with  the  same  per  cent  voltage  drop,  will  be  equal  to  the  square 
of  the  increase  in  voltage.  A  change  in  station  transformers, 
station  equipment  and  line  transformers  is  necessary.  The 
economical  advantage  can  be  studied  by  a  careful  consideration 
of  the  annual  costs  including,  of  course,  the  costs  of  making  the 
change. 

Another  method  is  to  change  from  a  single-phase  to  a  three- 
phase  system  for  the  main  lighting  circuits.  The  advantage 
gained  is  the  well-known  advantage  of  three-phase  over  single- 
phase.  Twice  the  load  can  be  carried  for  the  same  per  cent  power 
loss  and  per  cent  voltage  drop.  Balanced  against  this  advantage 
is  the  cost  of  making  station  changes,  of  installing  the  third 
conductor,  etc. 

A  combined  increase  in  voltage  and  a  change  from  single-  to 
three-phase  is  sometimes  used,  such  as  a  change,  from  a  2,300- 
volt,  single-phase  to  a  4,600-volt,  three-phase  system.  This 
method  has  the  advantage  of  obtaining  an  increase  in  capacity, 
due  to  both  changes  with  the  cost  of  making  the  change  very 
little  greater  than  for  either  change  alone. 

A  method  which  has  been  used  recently  in  a  number  of  places 
with  apparent  satisfaction  is  the  installation  of  a  four-wire,  three- 


LIGHTING  CIRCUITS  137 

phase  system,  with  grounded  neutral.  For  example,  if  2,300- 
volt,  single-phase  circuits  have  been  in  use,  the  change  is  made  to 
three-phase  with  a  voltage  of  2,300  volts  from  each  phase  to 
neutral.  Some  of  the  advantages  claimed  are: 

1.  The  reduction  in  voltage  drop  and  power  loss  due  to  three- 
phase  transmission  from  substation  to  feeding  points  or  branches. 

2.  The  reduction  in  total  number  of  conductors  due  to  the 
combination  of  three  former  single-phase  circuits  with  six  conduc- 
tors into  one  three-phase  circuit  with  four  conductors. 

3.  The  utilization  of  the  same  branch  single-phase  circuits  and 
distribution  transformers  without  change. 

4.  The  load  need  not  be  so  well  balanced  as  with  three-phase, 
three- wire  circui"'  'on  account  of  the  use  of  a  neutral  conductor. 
Also  the  ^hutting'  down  of  one  phase  will  not  cut  off  service  on  the 
other  pM'ses. 

5.  The   station   changes   are   somewhat   simpler  than  for   a 
straight  increase  in  voltage  or  change  to  a  three-phase,  three- wire 
system. 

6.  The  ability  to  carry  three-phase  loads  on  the  same  circuit 
if    desired. 

Some  disadvantages  noted  are  the  increase  in  voltage  from 
phase  wire  to  ground  on  single-phase  branches,  the  concentration 
of  fairly  large  loads  on  one  circuit,  and  the  fact  that  the  increase 
in  voltage  is  not  as  great  as  could  be  made  by  a  straight  doubling 
of  voltage. 

Another  method  which  has  also  been  tried  out  in  a  few  places 
is  that  of  extending  the  secondary  transmission  system.  Instead 
of  attempting  to  carry  the  load  all  out  of  main  substations,  on 
heavy  feeder  circuits,  at  primary  voltage,  small  automatic  or 
semi-automatic  substations  are  established  at  desirable  feeding 
points  and  these  are  supplied  through  high-voltage  lines  or 
cables.  The  primary  circuits  running  from  these  substations  are 
comparatively  short  and  a  corresponding  advantage  in  regulation 
is  accomplished. 

No  one  of  the  above  methods  of  increasing  the  capacity  of  the 
lighting  circuit  system  can  be  recommended  as  most  advan- 
tageous for  all  cases.  The  one  best  applicable  to  any  system 
must  be  chosen  by  a  careful  study  of  present  loads,  conditions 
in  the  substations  and  on  the  lines,  probable  future  loads,  the  cost 
and  practical  difficulties  of  making  the  change,  and  adaptability 
of  the  new  system  to  still  further  changes  when  necessity  de- 


138  ECONOMICS  OF  ELECTRICAL  DISTRIBUTION 

mands.  This  study  should  be  based  on  as  complete  a  determina- 
tion as  possible  of  annual  costs  before  and  after  the  change  and 
the  comparative  annual  costs  with  various  alternative  changes. 

Importance  of  Good  Service. — The  importance  of  good  service 
in  connection  with  lighting  circuits  should  be  emphasized.  It  is 
here  especially  that  a  desire  for  economy  should  not  lead  the 
engineer  to  practices  which  will  endanger  the  quality  or  con- 
tinuity of  the  service  rendered.  A  fluctuation  of  voltage  is  easily 
discernible  in  the  effects  on  lighting  and  gives  rise  to  many 
complaints.  An  interruption  of  service,  especially  if  for  any 
considerable  length  of  time,  discommodes  a  great  number  of 
customers,  sometimes  with  very  serious  consequences.  Economy 
is  always  desirable,  but  it  is  false  economy  to  save  a  few  dollars 
on  construction  and  thereby  lose  customers.  The  aim  should 
be  to  reduce  cost  to  a  minimum  which  is  consistent  with  service, 
at  least  as  good  as  that  to  which  the  customers  have  been 
accustomed. 


CHAPTER  XIII 
SECONDARY  DISTRIBUTION— SINGLE-PHASE 

STUDY    OF    MOST    ECONOMICAL    DESIGN    FOR    SECONDARIES — 
VOLTAGE    DROP — CONDUCTOR    SIZE — TRANSFORMER 
SIZE — LENGTH    OF    SECONDARY 

In  working  toward  the  efficient  and  economical  design  of  the 
central-station  system  as  a  whole  no  link  in  the  chain  connecting 
the  consumer  with  the  coal  pile  may  be  overlooked.  The  ulti- 
mate purpose  of  all  study  in  this  direction  is  to  enable  energy  to  be 
delivered  to  the  customer  at  the  least  possible  cost  per  unit, 
while  at  the  same  time  good  service  is  maintained.  To  this 
purpose  considerable  attention  has  been  paid  to  generating  plant, 
transmission  lines  and  substations  but  on  the  final  link  before 
reaching  the  customer — the  distribution  lines — the  tendency 
has  been  to  apply  ''rule  of  thumb"  methods  and  " experience" 
only  to  the  layouts.  When  it  is  considered  that,  even  in  a  well- 
designed  system,  the  investment  in  distribution  lines  will  often 
be  from  one-fifth  to  one-fourth  of  the  total  investment  on  the 
system,  and  that  the  energy  losses  on  these  lines  will  be  equal  to 
or  somewhat  more  than  one-half  of  the  total  loss  between  the 
generator  and  the  customer,  it  may  be  expected  that  a  study  of 
the  economical  design  of  distribution  lines  will  be  found  of  great 
profit. 

There  are  several  conditions  pertaining  to  the  secondary  system 
which  make  the  careful  layout  of  such  a  system  especially  impor- 
tant. The  number  of  transformer  installations  is  so  large  and  the 
lines  spread  over  so  great  an  area  that  constant  or  very  frequent 
inspection  is  impossible.  The  load  is  subject  to  irregular  in- 
creases. In  districts  which  are  newly  built  up,  new  services  are 
constantly  being  added.  In  old  districts,  new  appliances  are 
being  purchased  and  the  load  on  old  services  thereby  increased. 
On  this  account  any  design  must  be  made  to  cover  a  period  of 
years  and  the  increase  in  load  for  that  period  estimated  from 
past  experience.  On  the  other  hand,  care  must  be  taken  not  to 
install  too  much  capacity  and  thereby  increase  the  cost  beyond 
the  limits  of  economy.  The  problem  must  be  carefully  studied 
to  obtain  the  balance  between  low  cost  and  good  service  for  any 
particular  case. 

139 


140  ECONOMICS  OF  ELECTRICAL  DISTRIBUTION 

The  problems  of  secondary  distribution  economics  deal  chiefly 
with  the  wire  size  and  the  size  and  arrangement  of  transformers. 
The  voltage  is  usually  limited  to  a  small  range  of  values  by  past 
practice,  transformer  standards,  lamps,  motor  and  other  appli- 
ance standards,  etc.  The  layout  of  pole  locations,  while  requiring 
a  considerable  amount  of  engineering  skill  and  experience,  is 
usually  dependent  on  local  conditions,  the  arrangement  of  lot 
lines,  convenience  in  reaching  services,  probable  future  business, 
etc.  rather  than  on  purely  economic  considerations.  Of  course, 
many  local  problems  arise  in  the  layout  and  construction  of 
secondary  lines,  in  the  solution  of  which  economics  should  be 
considered.  Changes  in  type  of  construction  should  be  looked 
at  from  an  economic  view  point  as  well  as  from  that  of  the  me- 
chanical design  only. 

There  are  several  general  types  of  problems  relating  to  second- 
aries. They  might  be  classified  in  general  as  those  of : 

(a)  Single-phase  secondaries  in  cities  and  large  towns. 
(6)  Single-phase  secondaries  in  small  towns  and  country. 

(c)  Three-phase  secondaries  on  large  power  installations. 

(d)  Three-phase  secondaries  on  small  power  installations. 

In  this  chapter  the  problem  of  single-phase  secondaries  will 
be  discussed,  especially  in  reference  to  well  built  up  districts 
where  the  load  may  be  considered  to  have  practically  a  uniform 
distribution. 

Secondaries  for  Uniformly  Distributed  Load. — In  attacking 
such  a  problem  we  can  often  determine  from  tests  and  from  past 
experience  what  the  density  of  the  loading  is  and  how  it  will 
increase  for  some  years  in  advance.  We  are  usually  limited  to 
certain  stock  sizes  of  transformer  and  of  wire,  on  any  system,  due 
to  practical  considerations  of  manufacturing  and  stock  keeping. 
The  problem  then  is  to  determine  the  proper  combination  of  wire, 
transformer  and  transformer  spacing  in  order  to  give  good  condi- 
tions of  operation  and  also  to  show  the  least  cost  per  year  for  the 
load  densities  expected  during  the  period  of  time  under  considera- 
tion. It  is  clearly  understood  that  no  definite  rules  can  be 
established  which  will  fit  all  conditions.  The  variations  in  the 
problem  are  too  many.  The  most  that  can  be  done  is  to  furnish 
means  for  readily  discovering  the  limitations  of  any  problem 
and  of  proceeding  within  these  limitations  to  the  most  economical 
design. 


SECONDARY  DISTRIBUTION— SINGLE-PHASE  141 

The  study  has  been  carried  forward  from  three  different  angles. 
First,  from  the  theoretical;  second,  from  a  semi-practical,  that  is, 
by  adopting  certain  standards  and  studying  their  behavior; 
third,  from  a  purely  practical,  giving  the  designer  data  on  the 
costs  of  various  transformers  and  wire  sizes  under  the  conditions 
ordinarily  encountered  in  practice. 

In  all  this  discussion  it  has  been  assumed  that  the  loading  is 
such  that  it  may  be  considered  as  uniformly  distributed  along  the 
line.  The  unit  used  is  called  load  density,  given  in  kilowatts 
per  1,000  ft.  The  line  is  assumed  to  be  three- wire  secondary 
spaced  42  in.  between  outside  wires.  The  cost  of  right-of-way, 
poles,  crossarms  and  insulators  is  not  included  in  any  of  the  com- 
putations as  it  is  assumed  this  would  be  the  same  under  any  given 
condition.  Also  the  difference  in  length  of  primary  for  different 
transformer  spacings  is  not  considered.  In  actual  design  under 
known  conditions  a  correction  should  be  made  for  this.  The 
loading  conditions  are  taken  as  those  of  residence-lighting 
districts  although  the  same  methods  could  be  adapted  to  any 
other  conditions  of  loading  if  its  characteristics  were  known. 
Transformers  are  assumed  to  be  in  the  center  of  the  secondary 
served,  feeding  both  ways. 

DISCUSSION  OF  METHODS  USED  IN  DERIVING  EQUATIONS 
AND  THEIR  APPLICATION 

Theoretical. — In  the  theoretical  discussion  ideal  conditions 
are  assumed  which  will  rarely  if  ever  be  met  with  in  practice, 
but  it  can  be  shown  by  a  study  of  the  results  how  they  may  be 
applied  to  practical  conditions.  These  assumptions  are  that 
the  line  is  indefinite  in  length  so  that  the  transformers  may  be 
placed  at  any  exactly  determined  spacing  and  that  the  spacing 
will  change  with  the  load;  that  the  transformer  is  always  of  a 
size  just  equal  to  the  load  to  be  carried,  that  is,  equal  to  the 
load  density  at  peak  load  times  the  spacing;  that  the  wire  may 
be  of  any  cross  sectional  area  and  vary  with  the  load.  Such  a 
condition  could  only  be  obtained  in  a  case  where  the  load  showed 
only  seasonal  variations  and  no  yearly  increase.  However, 
in  practice  we  usually  design  for  a  certain  period  at  the  end  of 
which  it  is  assumed  the  load  density  will  be  a  certain  amount. 

The  general  method  has  been  to  obtain  an  expression  for  the 
annual  cost  per  1,000  ft.  of  line  and  to  determine  by  finding  the 


142  ECONOMICS  OF  ELECTRICAL  DISTRIBUTION 

first  derivative  and  setting  it  equal  to  zero,  the  condition  under 
which  this  annual  cost  is  a  minimum.  This  is  the  most 
economical  condition. 

Annual  Cost  of  Secondary  Distribution. — The  general  equation 
for  the  annual  cost  is  first  obtained  as  follows : 

Annual  cost  per  1,000  ft.  of  installation  =  Y 

Y  =  (Total  annual  charges  on  transformers  per  1,000 
ft.  of  line)  +  (Total  annual  charges  on 
line  per  1,000  ft.  of  secondary)  =  YT 
+  YL  (55) 

Where    YT  =   (-—)     (Purchase    price  +  cost    of    handling  + 

cost   of  installation  -f  cost   of   lightning 
arresters  and  equipment)  +  Cost  of  core 

and  copper  loss  +  cost  of  inspection  j  — ~— 

gT  =  Per  cent  interest  +  depreciation  +  taxes  on 

transformer. 
S  =  Spacing  of  transformers  in  feet. 

The  core  loss  is  practically  a  constant  quantity  for  24  hr.  per 
day  throughout  the  year.  The  copper  loss  on  the  other  hand 
depends  on  the  load.  If  the  characteristic  variation  of  this  load 
from  hour  to  hour,  day  to  day  and  month  to  month  is  known, 
the  average  loss  per  day  can  be  determined  in  terms  of  the  year's 
peak  load.  In  this  case  the  peak  load  is  assumed  to  be  just 
equal  to  the  capacity  of  the  transformer.  The  cost  of  energy 
at  the  transformer  must  also  be  carefully  determined.  The  cost 
for  copper  loss  will  be  considerably  higher  than  that  for  core  loss 
on  account  of  the  lower  load  factor.  The  sum  of  all  these  items 
makes  up  the  annual  cost  on  a  transformer. 

It  was  found  that  if  the  value  of  the  transfoimer  annual  cost 
is  plotted  against  the  transformer  size  that  the  curve  for  values 
between  0  and  25  kw.  may  be  approximated  by  a  straight  line 
of  the  formula  YT'  =  K\  +  K2T,  T  being  the  transformer  size 
and  KI  and  K2  constants  to  be  determined  for  any  particular 

combination  of  transformer  cost,  energy  cost,  etc.  (see  Fig.  36). 

i  nnn 
This  becomes  YT  =  -~-  (#1  +  K2T)  per  1,000  ft.,  where  S  is 

the  length  of  secondary  belonging  to  any  one  transformer  or  the 
distance  between  transformers  where  banked. 


SECOND AR  Y  DISTRIB  UTION— SINGLE-PHASE 


143 


Assuming  a  transformer  size  just  sufficient  to  carry  the  load, 
then 

T-T       S 
LD 


Where  LD  =  load  density  in  kw.  per  1,000  ft. 

mi.  TT-  1>000/T,  T,,       -L/jr>O  \  /!-r>\ 

Then  YT  =  _y  (*.  +  ^,  j  (56) 


50 


I30 

c 


Transformers  assumed  to  be  fully  loaded  a  f 
yearly  peak  load. 


Curve  of  total  cost  approximates  equation 
YT'=fr,  +  KzT 


)  s_  10  15  ao 

Ske  of  Transformers  in  KW=T 

FIG.  36. — Annual  charges  on  transformers. 


25 


The  annual  cost  on  the  line  includes  interest,  depreciation  and 
taxes  on  the  investment  cost  of  the  wire  in  place,  including 
purchase  price  and  cost  of  installation,  also  the  cost  of  annual 
energy  loss  due  to  resistance.  The  copper  loss  is  arrived  at  by 
the  same  method  as  the  copper  loss  on  the  transformer,  that  is, 
by  use  of  the  equivalent  average  number  of  hours  per  day  at 
full  load  or  equivalent  hours. 


144  ECONOMICS  OF  ELECTRICAL  DISTRIBUTION 

YL  =  Investment  cost  of  material  per  1,000  ft.  of  line  +  instal- 
lation charges  per  1,000  ft.  of  line  H — ^ —  X  cost  of 
copper  loss  in  secondary. 

YL  =  ^(3  X  1,000  X  w  X  Ccu  +  Csr)  +  ^^l  2  (^)  *  X  j  X 

fX2X<X365XI§o]        ^ 
Where  gL  =  per  cent  interest  +  depreciation  +  taxes  on  line, 

w  =  weight  of  insulated  wire  in  pounds  per  foot, 
Ccu  =  cost  of  insulated  wire  per  pound, 
Csr  =  cost  of  stringing  1,000  ft.  of  line, 
/  =  total  current  in  secondary  at  transformer, 
p  =  resistivity  of  wire  per  mil  foot, 
Cez  =  cost  of  copper  loss  in  secondary  per  kilowatt-hour, 
t  =  equivalent  hours  per  day  which  yearly  peak  load 
should  continue  in  order  to  give  an  PR  loss  equal 
to  the  total  actual  PR  loss  for  the  year, 
A  =  cross-sectional  area 'of  wire  in  circular  mils, 
E  =  voltage  between  outside  wires  of  secondary, 
Cos  0  =  power  factor  of  load. 
j   =     LDS 

E  cos  0 
and 
v         SL 


(3,000  wCcu  +  Ssr) 


(57) 


The  total  annual  cost  per  1,000  ft.  of  installation  is  now 
obtained  by  adding  these  two  quantities,  annual  cost  of  trans- 
formers and  annual  cost  on  line,  and  the  equation  obtained  as 
shown  below. 


Then  Y  -  (*  +  )  +  ^  (3,000  WCe,  + 


Equation  58  gives  the  total  annual  cost  per  1,000  ft.  of  instal- 
lation as  a  function  of  the  spacing  and  load  density. 

Most  Economical  Voltage  Drop.  —  One  of  the  most  important 
controlling  factors  in  determining  the  length  of  a  secondary  or  the 
spacing  of  transformers  is  the  maximum  allowable  voltage  drop. 


SECONDARY  DISTRIBUTION— SINGLE-PHASE  145 

It  has  usually  been  considered  that  the  most  economical  condition 
of  operation  would  be  with  a  voltage  drop  higher  than  would  be 
allowable  for  good  service.  In  our  case,  3  per  cent  drop  has  been 
considered  the  limiting  value,  as  luminosity  curves  for  Mazda 
lamps  show  a  reduction  as  high  as  18  per  cent  with  5  per  cent 
voltage  drop  while  3  per  cent  shows  over  10  per  cent  reduction. 
Considering  the  voltage  loss  in  the  service  drops,  which  cannot  be 
figured  closely  on  account  of  variable  conditions  and  the  fact  that 
the  load  is  not  absolutely  uniformly  distributed,  3  per  cent  is 
considered  the  highest  value  commensurate  with  good  operation. 
It  must  be  determined,  then,  if  under  certain  conditions,  a  smaller 
voltage  drop  than  this  will  be  more  economical. 

In  order  to  obtain  the  most  economical  per  cent  voltage  drop  it 
is  necessary  to  obtain  F  as  a  function  of  the  per  cent  voltage  drop. 

This  is  done  as  follows: 

W  =  total  load  on  secondary  in  watts, 
V  =  voltage   drop   on  secondary  in  per  cent  of  de- 
livered voltage, 

B  =  constant  relation  between  per  cent  voltage  drop 
and  the  per  cent  power  loss. 

A  F9*7' 
Then  W  =  LDS  =  ££ 

Whence  8 


300  BS 
E 


17.32 

Substituting  this  value  for  S  in  Eq.  58, 

Then  F  =  1?°°°  Kl—  +  K2LD  +  ^  (3,000  w  Ccu  +  Car)  + 


17.32 

#2       AV 


r  r  T  2nfr     1 

60.83   ,"P     -&\ 
,D  I  AE2  cos  20J 


(17.32)2  BL, 

=  17,320  ^  JP^p  V  ~y*  +  0.2028  P*C*L°  V 
E   \  A  B  cos2  6 

+  K2LD  +  ^  (3,000  X  Ccu  +  Csr)    (59) 


The  most  economical  per  cent  voltage  drop  is  obtained  when 
the  first  derivative  of  Y  with  respect  to  V  equals  0 
dY 

W  ' 

=  -y2  =  17,320  K,-^^  Vee-H+  0.2028  J^j^  =0 
10 


146 


ECONOMICS  OF  ELECTRICAL  DISTRIBUTION 


Solving  for  Vec 


Vec  =  1,223 


r  cos2  0 


(60) 


Equation  60  gives  the  most  economical  per  cent  voltage  drop 
as  a  function  of  the  load  density. 

By  assuming  values  for  the  constants  to  fit  particular  condi- 
tions this  expression  for  V  can  be  plotted  against  load  density 
for  various  standard  wire  sizes.  These  curves  show  that,  as 
load  density  increases,  the  most  economical  voltage  drop  de- 
creases and,  under  the  conditions  assumed  in  the  curves  here 
plotted,  the  most  economical  voltage  drop  falls  below  3  per  cent 
at  load  densities  which  are  often  encountered  with  such  loads 
(see  Fig.  37). 


|  . 

Load  uniformly  distributed-  Transformer  size  equal  fv  load 


10  15  20  25 

Load  Density  in  Kw.  per  1000  feet  = 

FIG.  37. — Most    economical   voltage    drop   in   per    cent   of   delivered    voltage. 


Most  Economical  Transformer  Spacing.  —  In  order  to  obtain 
the  most  economical  length  of  secondary  or  spacing  of  trans- 
formers it  is  necessary  to  have  Y  as  a  function  of  S.  This  is 
obtained  from  Eq.  58. 


(60.83 


AE2  cos  20 


SECONDARY  DISTRIBUTION—  SINGLE-PHASE  147 

The  most  economical  spacing  is  obtained  when  the  first  deriv- 
ative of  V  with  respect  to  S  equals  0  — 


Solving  for  Sec 

A*  . 


Equation  61  gives  the  most  economical  spacing  of  transformers 
as  a  function  of  the  load  density. 

It  is  necessary  to  limit  the  range  of  application  of  Eq.  61  to 
conditions  where  the  voltage  drop  is  less  than  3  per  cent.  A 
second  equation  must  be  developed  for  3  per  cent  drop  to  apply 
where  the  most  economical  drop  would  be  greater  than  3  per 
cent.  Practical  considerations  limit  the  drop  to  that  value. 

From  above 

S  = 


300  BLD 

Using  V  =  3  per  cent 


=  Transformer  spacing  for  3  per  cent  drop. 
Then,  summarizing, 

H 


which  is  general  for  all  cases.  If  now  the  constants  are  evaluated 
these  curves  may  be  plotted  for  various  sizes  of  wire,  using,  for 
any  particular  load  density,  the  equation  which  shows  the 
shortest  spacing.  We  obtain  the  set  of  curves,  Fig.  38,  giving 
the  transformer  spacing  which  will  give,  with  any  wire  size,  the 
greatest  economy,  providing  good  operating  conditions  are 
maintained  by  having  no  voltage  drop  greater  than  3  per  cent. 
Most  Economical  Transformer  Size.  —  It  is  a  simple  matter 
with  this  data  at  hand  to  derive  the  curves  showing  the  most 
economical  transformer  size  for  any  load  density,  providing  the 
transformers  are  spaced  most  economically.  Since  it  was 
assumed  in  the  beginning  that  the  transformer  would  be  just 

S 

large  enough  to  carry  the  load,  Tec  =  LD  r7^  (Eq.  64)  where  Sec 

1,000 


148 


ECONOMICS  OF  ELECTRICAL  DISTRIBUTION 


is  the  value  taken  from  the  curves  for  most  economical  spacing. 
This  is  the  most  economical  size  for  any  load  since  the  annual 
charges  on  the  investment  represented  by  the  transformer  is  a 


6000 


5000 


Load  uniformly  distributed-  Trans  former  size  jusf  equal  rofoad. 


15  20  25  30 

Load  Density  in  Kw.  per  1000  feet  =  LO 


40 


45 


FIG.  38. — Most  economical  spacing  of  transformers   (limited  by  a  maximum 
allowable  voltage  drop  of  3  per  cent). 


15  20  25  JO 

Load  Densiiy  in  Kw.per  1000  feel-  =  L0 


FIG.  39.  —  Most  economical  transformer  size  (being  just  equal  to  the  load  at  the 
most  economical  spacing). 


much  greater  proportion  of  the  total  annual  charge  than  the  cost 
of  energy  losses.  Therefore  the  use  of  a  larger  transformer,  even 
though  under-loaded,  would  be  more  costly  (see  Fig.  39). 


SECONDARY  DISTRIBUTION— SINGLE-PHASE 


149 


Most  Economical  Wire  Size. — It  is  now  possible  to  attack  the 
problem  of  the  most  economical  size  of  wire  for  any  load  density. 
We  will  assume  that  it  is  feasible  to  use  the  most  economical 
transformer  size  at  its  most  economical  spacing  for  any  load 
density,  modified  by  the  limiting  3  per  cent  voltage  drop  require- 
ment. Then  if  we  substitute  in  our  original  equation  (Eq.  58) 
the  expressions  for  S  used  in  plotting  the  curves  for  most  eco- 
nomical spacing  and  for  spacing  limited  by  3  per  cent  drop  in 
voltage,  we  obtain  two  expressions  for  the  annual  cost  per  1,000 
ft.  in  terms  of  load  density  and  cross-sectional  area  of  wire  (A). 
It  is  necessary  to  introduce  two  approximations.  The  weight 
per  foot  of  wire  (w)  enters  the  equation,  also  the  quantity  B  which 


1.12 
LOB 
CO  1.04 
1.00 

B=  Re 

a 

tot 

3°; 

Cur 

atlonbet'w 
nd  percen 

jes  taken 
y  voltage 
ve  approjtit 

sen  percan 
f  power  to 
it  42  "spa 

t-  voltage  c 
ss  in  alim 
'ing  and 

'rop. 

/ 

/ 

50 

nates  equ 

i+i  on  0=A 

f»fs* 
/ 

/ 

3t 

/ 

Y 

^ 

5  3  S  i 

ight  of  T.  &.W.P.  Wire  in  Ibs.per  10 

/ 

/ 

^ 

^ 

-^ 

A 

^ 

^ 

Weight  o 
per  100  f1 
Curve  app 

"insulated 
T.B.W.P.w 
roximates 
KWfr 

too    (p 

wire  in  po 
re. 
equation 

unds 

0* 

^ 

-"•" 

W= 

er  foot)  - 

WJ 

>°0 

)             10,000         20,000        30,000        40,000         50,000        60,000         10,000         80,000        90,0 

Site  of  Wire  inCircwIar  Mils- A 

FIG.  40. 


is  the  constant  relation  between  per  cent  voltage  drop  and  per 
cent  power  loss  for  any  size  of  wire.  It  is  found  by  plotting  values 
of  w  for  standard  sizes  of  wire  of  the  range  of  sizes  which  would  be 
used  in  secondaries  that  the  expression  W  =  Ks  +  K4A  is  a  very 
close  approximation,  K3  and  K^  being  constants  (see  Fig.  40). 
Also  it  is  found  that  the  value  of  B  for  any  size  of  wire  may  be 
approximated  very  closely  by  the  straight  line  function  B  = 
K$  +  K&A,  where  K5  and  K&  are  constants  (see  Fig.  40).  These 
must  be  derived  from  the  particular  values  of  B  which  apply 
to  the  conditions  being  studied  since  these  values  vary  for  differ- 
ent spacings  between  wires.  Substituting  these  expressions  in 
the  equations  referred  to  above  we  obtain  the  two  general 


150  ECONOMICS  OF  ELECTRICAL  DISTRIBUTION 

expressions  for  annual  cost  per  1,000  ft.  in  terms  of  wire  size  for 
maximum  economy  of  transformer  spacing  and  for  3  per  cent 
voltage  drop,  Eqs.  65  and  67.  These  are  differentiated  with 
respect  to  A  and  the  Eqs.  66  and  68  are  obtained  between 
the  most  economical  wire  size  and  the  load  density  for  most 
economical  spacing  and  for  3  per  cent  voltage  drop. 
Substituting  the  value  Sec  (Eq.  61)  for  S  in  Eq.  58 

+  C8r) 


PtCe2 
+  60.83 


If  w  =  K*  "  (see  curve  40). 


The  equation  for  annual  costs  per  1,000  ft.  becomes 


+*** 

Simplifying 


30(^3  +  K*A)CCU  +  Car         (65) 


Equation  65  gives  the  annual  cost  per  1,000  ft.  of  line,  using  the 
most  economical  spacing  of  transformers. 

The  most  economical  cross-section  of  wire  is  obtained  when  the 
first  derivative  of  Y  with  respect  to  A  equals  0  or 
dY 


=    _       V  744  f)  !^    ^  .      _^     ,     J/L_    y.  on/?-  /nr 

3  X          °  L#2cos20J      D      ec        h  100  X 
Solving  for  Aec 

154      TKJptC^     „  (    . 

Aec  =  LD 


Equation  66  gives  the  most  economical  cross-section  of  wire 
using  the  most  economical  spacing  of  transformers. 

It  is  necessary  to  limit  the  application  of  Eq.  66  to  less  than 
3  per  cent  voltage  drop  and  develop  the  equation  for  most 
economical  wire  size  with  3  per  cent  drop.  This  is  done  as 
follows: 


SECONDARY  DISTRIBUTION— SINGLE-PHASE  151 

From  Eq.  62,  the  spacing  which  will  give  a  3  per  cent  voltage 
drop  is, 

E  rr 

=  W\BTD 

B  may  be  expressed  as  a  function  of  A  as  follows: 
B  =  K$  +  K&A  (see  curve  40). 

S  =  fQ 

Substituting  the  value  of  S  in  Eq.  58,  the  expression  for  annual 
costs  per  1,000  ft.  of  line  (the  spacing  being  limited  for  a  3  per 
cent  voltage  drop)  becomes 
v  1,000X1 

4  3  per  cent  —  — 


10 

CU    +  Csr] 


AE2  cos2  6  "  100  (K6  +  KtA)LD 
Simplifying 


3  per  cent   ~  ™ 

cu  +  Csr] 


+  0.6083  -    2er  ^      (67) 
cos2  e(K$  + 


.  Equation  67  gives  annual  cost  per  1,000  ft.  of  line  using  a 

spacing  which  limits  the  voltage  drop  to  3  per  cent  at  full  load. 

The  most  economical  cross-sectional  area  is  obtained  when  the 

first  derivative  of  ¥3 percent  with  respect  to  A  is  equal  to  0. 

dY  3  per  cent.    _   ^ 

dA 

ptCezLD 


-  .6083 
From  which 

T      14 

'   I    i-r^'" 


cu     (68) 


152 


ECONOMICS  OF  ELECTRICAL  DISTRIBUTION 


Equation  68  gives  the  most  economical  cross-sectional  area  of 
wire  when  the  spacing  is  limited  by  a  3  per  cent  voltage  drop 

The  constants  were  evaluated  and  these  curves  plotted,  the 
3  per  cent  curve  for  low  load  densities  and  the  maximum 
economy  curve  for  high  loading.  They  furnish  a  graphic  repre- 
sentation of  the  most  economical  size  of  wire  to  use  under  any 
load  density  providing  ideal  conditions  obtain  in  the  way  of 
transformer  size  and  spacing.  See  Fig.  41. 


s«,ww 

II    innnn 

Tnmsfe 
econor 

rmer  she  assumed 
•jical  spacing  /imifea 

iu&teeyual 

+o  load  at 
•age  drop 

most 
#4w» 

IE 

^f 

j^jgSSJ 

Cross-  Sectional  Area  in  Cl'rcolar  Mils 

•_1_J_JLJ 

** 

*** 

^^~ 

* 

6  WIRE 

^ 

^ 

A 

1 

3                 5                10               I 

70               25                30                } 

b              40               4 

Load  Density  in  Kw.per  1000ft  =  LO 

FIG.  41. — Most  economical  wire  size. 

Purpose  of  Theoretical  Curves. — At  first  glance  it  may  appear 
as  if  these  curves,  being  obtained  on  the  basis  of  such  theoretical 
assumptions,  could  have  very  little  practical  value.  However, 
when  attacking  a  practical  problem  of  this  nature  the  data  from 
these  curves  may  be  used  as  the  basis  upon  which  to  start  the 
calculations  of  annual  costs  under  operating  conditions.  If, 
for  example,  the  present  load  density  and  the  load  density  which 
is  to  be  expected  at  some  certain  future  time  are  known,  by  going 
to  the  theoretical  curves  we  may  determine  (a)  whether  the  volt- 
age drop  is  to  be  limited  by  the  3  per  cent  maximum,  (6)  what 
would  be  the  most  economical  conditions  of  transformer  size  and 
spacing  for  present  operation  and  for  operation  at  that  future  time, 
and  (c)  what  standard  size  of  wire  will  be  most  economical  over 
the  period.  The  curve  for  the  most  economical  wire  size  covers, 
for  each  standard  size,  such  a  range  of  load  densities  that  we 
should  be  able  at  once  to  select  our  wire  size  without  further 
computation.  Having  determined  this  and  knowing  what 
stock  sizes  of  transformers  and  practical  spacings  come  the 
nearest  to  fitting  the  ideal  conditions  over  the  period  under 


SECOND AR  Y  DISTRIB  UTION— SINGLE-PHASE 


153 


consideration,  we  can  then  investigate,  as  will  be  shown  later, 
the  comparative  economy  of  such  various  methods  of  installa- 
tion as  could  be  used  in  this  particular  case.  In  other  words, 


100 


SO 


10 


t 

£60 
o 

o 
o 


c 
'\n 
8,40 


iO 


Trans  forme  resizes  and  Spacings  assumed  to  be 
those  most  theoretically  economical  limited 
by  3  %  maximum  voltage  drop. 

Annual  cost  includes  line  and  transformers. 


0  5  10  15  ZO  £5  30 

Load  Density  in  Kilowatts  per  1000  feet 

FIG.  42. — Curves  showing  comparative  economy  of  various  wire  sizes  in  sec- 
ondary installations. 


these  theoretical  curves  give  certain  limitations  on  which  we  may 
proceed  to  further  more  practical  investigation. 

Semi-practical. — In  order  to  present  our  results  in  a  little 
more  concrete  and  practical  form  and  to  show  the  exact  com- 
parative economy  between  various  types  of  installation,  espe- 


154  ECONOMICS  OF  ELECTRICAL  DISTRIBUTION 

cially  with  respect  to  the  size  of  wire  to  be  used,  a  series  of  curves 
were  developed  showing  the  exact  annual  cost  under  various 
conditions.  These  are  called  the  semi-practical  curves  (Figs. 
42  and  43). 

Annual  Cost  of  Standard  Wire  Sizes  Working  under  Ideal 
Conditions. — The  first  condition  was  assumed  to  be  that  in 
which  the  most  economical  size  of  transformer  could  be  used, 
spaced  the  most  economically  or,  where  necessary,  for  3  per  cent 
maximum  voltage  drop.  A  curve  was  plotted  for  each  of  the 
three  standard  sizes  of  wire,  No.  6,  No.  4  and  No.  2,  showing  the 
annual  cost  at  various  load  densities  (see  Fig.  42).  This  is,  in 
reality,  simply  plotting  Eqs.  65  and  67  as  developed  above. 

Annual  Costs  per  1,000  ft.  of  Installation  for  Any  Combination 
of  Standard  Sizes  of  Wire  and  Transformers. — As  the  next  step 
in  proceeding  from  the  general  problem  to  the  concrete  example 
various  combinations  of  standard  sizes  of  transformers  with 
standard  sizes  of  wire  were  assumed  and  curves  developed 
showing  the  annual  cost  of  each  of  these  combinations  at  various 
load  densities.  The  transformer  spacing  was  still  assumed  to  be 
always  the  theoretically  best  spacing  for  each  particular  load. 
This  enables  us  to  compare  for  example  the  economy  of  a  10-kw, 
transformer  and  No.  4  wire  with  that  of  a  15-kw.  and  No.  6 
wire  at  any  load  density. 

The  method  of  developing  these  curves  has  some  points  of 
interest  although  the  equations  are  merely  variations  of  our 
general  equation  for  annual  cost  per  1,000  ft.  It  is  seen  that  for 
any  size  of  transformer,  as  the  load  density  increases  a  certain 
point  is  reached  where  the  spacing  is  no  longer  governed  by  the 
allowable  voltage  drop  but  by  the  size  of  the  transformer  itself. 
Hence  each  curve  will  consist  of  two  parts,  the  lower  where  the 
voltage  drop  governs  the  spacing,  and  excess  transformer  capacity 
is  provided,  the  upper  where  the  transformer  size  governs  the 
spacing  and  the  voltage  drop  is  less  than  the  allowable.  The 
total  annual  cost  is  made  up  of  five  items : 

1.  Transformer  core  loss. 

2.  Transformer  copper  loss. 

3.  Copper  loss  on  the  line  itself. 

4.  Fixed  charges  on  the  transformer  (interest,  depreciation, 
taxes,  inspecting,  tests,  etc.). 

5.  Fixed  charges  on  the  line.     (Interest  and  depreciation.) 
Each  of  these  five  elements  was  analyzed  as  to  constants  and 


SECONDARY  DISTRIBUTION— SINGLE-PHASE  155 

variables,  considering  the  load  density  LD  as  the  chief  variable, 
and  the  transformer  and  wire  sizes  constant  for  any  given  condi- 
tion. It  was  found  that  the  equations  took  the  following  form : 

Y  =  KSLDH  +  K9LD  +  (K7  +  Klo)LD*  +  Kn 
when  the  voltage  drop  and  wire  size  governs,  and 

Y  =  (K12  +  #13  +  K15)LD  +  (Ku  +  Xu) 
when  transformer  size  governs. 

The  first  is  an  equation  of  a  third  degree  curve  in  LD*  breaking 
into  a  straight  line  (the  second  equation)  at  the  critical  point 
where  the  spacing  for  3  per  cent  drop  fully  loads  the  transformer. 
The  equation  for  each  constant  was  then  developed  and  evaluated 
for  each  combination  of  wire  and  transformer.  The  expressions 
for  costs  here  given  differ  from  those  given  in  the  theoretical 
discussion  in  that  here  actual  stock  sizes  of  transformers  and 
standard  wire  sizes  are  used.  The  results  were  then  plotted  as 
shown  in  Fig.  43.  The  derivation  of  these  curves  is  a  good 
example  of  the  method  of  developing  a  general  curve  by  the 
use  of  symbols  for  all  constants  and  then  evaluating  these  sym- 
bols to  fit  a  given  condition. 

The  derivation  follows: 

The  load  possible  on  a  given  wire  with  a  3  per  cent  drop  is 
given  by  the  formula: 

3AE*  AE2 


W  = 

B  X  300  X  S      WOBS 

W  =  LDS  and  / 


LDS 


E  cos  0 

77!  I        ~l 

S  =  Tft\/T>7-  wnen  limited  by  voltage  drop  (see  Eq.  62). 

JLU  \  JjLjj) 

L  S 
The  total  load  on  the  transformer  =      D 


1  j 

1  000  T 
S  =     '  T     -  when  limited  by  transformer  capacity. 

LiD 

The  following  items  enter  into  the  total  annual  charges  per 
1,000  ft.  of  installation. 

(a)  Transformer  Core  Loss. — This  is  assumed  constant  for  a 
given  transformer  for  all  loads. 

Core  loss  =  constant. 

1  000 
Annual    charge    per    1,000    ft.  =  — ^—  X  CeJ  X  24  X  365  X 

core  loss 
1,000 


156  ECONOMICS  OF  ELECTRICAL  DISTRIBUTION 

Cel  X  24  X  365  X  IPX  core  loss  v 

-  X  Lj>H  (limited  by  voltage 
drop) 


or 


ei  X  24  X  365  X  core  loss  T 

1  000  T  —  ~~       (lirmted  by  transformer  size) 


Whence, 

«.         d  X  24  X  365  X  10  X  core  loss 


„      _     ei 


X  24  X  365  X  core  loss 


i,ooor 

Cei  =  cost  of  core  loss  per  kilowatt-hour 
Core  loss  in  watts. 

(b)  Transformer  Copper  Loss.  — 

1,000  /  365  X  t 

/  CdX 


_  1,000  /  L,,^8  365  X 

s  \^COS2^^         ^ooo 

LD*RTCe2  X  365&S 
J5/2  cos2  6. 

RTCe2  X  365  X  t^jj 

cos2!  ----  Lz>^  (limited  by  voltage  drop) 


X  365  X  1,000  X  t  X  T  , 

~E2  cos2(9  ~     D  (limited  by   trans- 

former size) 


Where 


ez  3652  \/-^ 

TV-  _  \Z> 


cos2^ 
365,000*  X  T 


(c)  Secondary  Copper  Loss.  — 


SECONDARY  DISTRIBUTION— SINGLE-PHASE  157 


Secondary  copper  loss  =  ^P  2  (^)    XjXgX2X*X 

365  X 


1,000 
365Ce2S2 

by  volt- 


e  ,    ,       ..    ^    .  ,  .  .    , 

or  =  ^-sj 2~Z1 —  (when  limited  by  transformer  size) 

=  KL, 

Where 


cos20£600 


(d)  Fixed  Charges  on  Transformer.  —  Constant  for  any  size  of 
transformer. 

l,000r  gT  (Transformer  cost  +  lightning  arrester  +1 
S     LlOO     cost  of  installation)  -f  inspection  costJ 

104   [         (Transformer   cost  +  lightning   ar- 
rester  +  cost  of  installation)  +  in- 
spection  cost 
(when  limited  by  voltage  drop) 
=  X10LD« 

1  1  gT  (Transformer  cost  +  lightning  arrester  +1 
T   100  cost  of  installation)   +  inspection  cost  I  LD 
(when  limited  by  transformer  size) 

=  Ki$LD 
Where 

„  104   r  gT   (Transformer  cost  +  lightning  arrester! 

/J  L  1  00    +  cost  of  installation)  +  inspection  cost  J 


-^  If  gT   (Transformer    cost  +  lightning   arrester  +~] 

Llb       rLlOO    cost  of  installation)  +  inspection  cost         J 
(e)  Fixed  Charges  on  Wire. 

CU  -f  Ctr) 


158 


ECONOMICS  OF  ELECTRICAL  DISTRIBUTION 


(/)   Total  Annual  Cost. — The  total  yearly  cost  per  1,000  ft.  of 
line  is  the  summation  of  these  five  items. 


too 


90 


80 


!TO 


Load  uniformly  distributed./   / 
Voltage  drop  most        /    I      / 
economical  maximum  3%      '      / 


'jm 


GENERAL  FORMULAE 

#  f  As  £0  y-  f/f7  ^;  ^ 
(Where  3%  dropgoverns  transformer  spacing) 


(Where  transformer  she  gove  rns 
trans  former  spacing) 


0  5  10  15  20  ZS  JO 

Load  Density  m  KW.per  IOOOfeet«L0 

FIG.  43. — Curves   showing   comparative   economy   of   various   combinations   of 
secondary  installations. 


Hence 
Ys  per  cent 


/ 


'     -j- 


(when  limited  by  the  voltage  drop) 
Y  =  KizLiD  +  KiSLD  +  KU  +  Ki$LD 
-  (K12  +  X13  +  ^i5)Lc  +  (X14  + 

limited  by  transformer  size) 
(see  curve  Fig.  43). 


u    (69) 


(when 
(70) 


SECONDARY  DISTRIBUTION— SINGLE-PHASE 


159 


Purpose  of  Semi-practical  Curves. — These  semi-practical  curves, 
although  reducing  the  variable  elements,  still  retain  enough  of 
the  ideal  condition  so  that  they  cannot  be  used  as  an  absolute 
criterion  but  merely  as  a  guide.  They  do  show  however  con- 
cretely the  relative  economy  of  the  various  standard  sizes  of  wire 
when  used  under  the  most  favorable  conditions  and  this  may  be 
taken  as  a  guide  to  their  comparative  behavior  under  all  condi- 
tions. The  second  set  of  curves  also  shows  concretely  the 
relative  economy  of  the  various  transformer  sizes  with  any  one 


eooo 


5000 


I  I      I        I 

J  Wire  Secondaries  =  244/122  Volts  of  cusfome 
mast  economical  transformer  spacing 
limited  bu  3%  voltage  drop      \ 
Power  Factor  95%  }    ' 


L  oad  uniformly  distributed  42  inch  spacing 
between  outside  wires. 


FIG.  44.- — Load  curves  for  secondaries. 


size  of  wire  as  well  as  the  relative  economy  of  various  sizes  of  wire 
with  any  size  of  transformer.  This  comparison  of  economy  is 
valuable  in  showing  the  exact  amount  which  the  annual  cost  of  one 
installation  is  greater  or  less  than  another.  It  often  occurs  that 
where  the  difference  in  cost  is  not  great,  other  advantages  are 
sufficient  to  more  than  offset  it  and  lead  to  the  choice  of  the  more 
costly.  The  spacing  of  transformers  is  here  considered  to  be  the 
maximum  allowable  throughout,  with  the  transformer  carrying 
its  maximum  allowable  load.  This  limits  the  general  appli- 
cation of  these  curves  in  practice  and  hence  like  the  first  series 
they  are  chiefly  useful  in  establishing  limits  and  as  a  basis  for  the 
design. 


160 


ECONOMICS  OF  ELECTRICAL  DISTRIBUTION 


Practical. — We  now  come  to  the  development  of  the  curves 
which  the  designer  may  use  in  testing  the  economy  of  any  design 
and  thereby  choose  the  most  economical  from  several  alter- 
natives. Here  no  "most  economical"  conditions  need  be 
assumed.  The  curves  simply  represent  annual  costs  as  they 
occur  under  any  condition  which  may  be  encountered. 

Load  Curves  for  Secondaries. — The  first  curve  is  a  development 
from  the  two  theoretical  curves,  the  most  economical  transformer 
spacing  and  most  economical  transformer  size. 


15  20  15  30 

Load  Densi-ty  in  KW.per  1000  ft-.  =  L, 


FIG.  45a.  —  Line  cost  curves.     Annual  cost  per  1,000  ft.  of  3  No.  6  secondaries  in- 
cluding fixed  charges  and  cost  of  lost  energy. 


From  Eq.  63 

Sec  =  2.02 


By  plotting  the  transformer  size  against  the  spacing  we  obtain 
for  each  size  of  wire  a  curve  showing  the  most  economical  spacing 
or  the  spacing  limited  by  3  per  cent  voltage  drop  for  any  total 
load  on  the  transformer  (see  Fig.  44).  By  drawing  diagonal 
lines,  one  for  each  load  density  desired,  we  can  now  show  for  any 
particular  load  density,  the  maximum  economical  spacing,  and 
the  minimum  transformer  size  with  that  density  and  spacing. 
This  curve  merely  simplifies  the  former  two  and  serves  the  same 
purpose,  not  introducing  any  new  principles.  It  is  evident  that 
any  point  below  the  curve  will  indicate  a  drop  less  than  the  value 
used  on  the  curve.  This  curve  is  of  use  in  determining  what 


SECOND AR  Y  DISTRIB  UTION— SINGLE-PHASE 


161 


alternative  designs  may  be  feasible  with  any  load  and  standard 
equipment  and  what  changes  may  be  made  to  care  for  an  increase. 
Line  Cost  Curves. — The  equation  for  annual  charges  on  the 
line  (Eq.  57)  is  next  developed  numerically. 

YL  -         (3,000,0,  +  <?„)+  60. 


35 


40 


5  10  15  20  25  30 

Load  Density  in  KW.per  1000ft  =  L  D 

FIG.  456. — Line  cost  curves.     Annual  cost  per  1,000  ft.  of  3  No.  4  secondaries  in- 
cluding fixed  charges  and  cost  of  lost  energy. 


10 


35 


40 


45 


15  20  25  30 

Load  Density  in  KW.per  1000 ft.  =  L0 

PIG.  45c. — Line  cost  curves.     Annual  cost  per  1,000  ft.  of  3  No.  2  secondaries  in- 
cluding fixed  charges  and  cost  of  lost  energy. 

All  constants  were  evaluated  and  a  curve  plotted  for  each 
desired  spacing — 100-ft.  intervals  were  used — showing  the 
annual  charges  per  1,000  ft.  in  terms  of  the  load  density  for  each 

standard  size  of  wire  (see  Fig.  45,  a,  b,  c). 
11 


162 


ECONOMICS  OF  ELECTRICAL  DISTRIBUTION 


Transformer  Cost  Curves. — The  third  set  of  curves  shows  the 
annual  cost  on  the  transformer  for  any  loading  (see  Fig.  46). 


\ 

c 

i 

•  c 

frto/« 
7/"^r^ 

/7C/W^ 

?fe/7e 

innut 
isforn 
PS  //xe 
'SK/^ 

Jftfel 

?/C05 

7er3  ^ 
d  cha 
vses 

•««£ 

t  on\ 
nder 
rges 

'anoL 
varioi. 
znd  c 

s  siz 
jsloa 
?sf 

?s 

*y 

+_     3500 

1 
g   3250 

0 

fc_  3000 
o    2150 

\ 

\ 

^ 

' 

/ 

i    „ 

—         —  ' 

___Ji 

-^ 

/ 

o  i  |;?U 

I  2500 
1 

£.     ™CO 

\          * 

\                    "    "0 

\              J)  JV 

\                   1- 

\            — 

' 

> 

f 

r 

Scale  for  Reducing  "Mai  Annual  Cost 
ooi-J  o  ?3  G;  J  g  ? 

5  §  S  g  s  §  S  g  * 

\       0 

U.        c 

w      '.Z  An 

1  S 

£* 

^ 

#^ 

/ 

|1 

\K      C   -  n 

-U^30 

// 

^J 

\    ^ 
\ 

jr 

V 

\ 

0 

J 

10       15       tQ      25      30       35      40 


FIG.  46.  —  Transformer  cost  curves. 


Annual  cost  of  the  transformer  = 

QT  (cost  of  transformer  +  cost  of  lightning  arrester  + 


y 


100 


cos^  °f  installation)  +  inspection 
+  cost  of  core  loss 
4-  cost  of  copper  loss 


=  ~~;  (cost  of  transformer  +  cost  of  lightning  arresters. 

1UU 

+  cost  of  installation) 
+  cost  of  inspection 

+  C.,  X  365  X  24  X 


E2  cos2  0 


-RT  X  365  X  t  X 


1,000 


(71) 


SECONDARY  DISTRIBUTION— SINGLE-PHASE  163 

The  equation  for  each  of  the  standard  sizes  of  transformers 
2,  5,  10,  15  and  25  was  developed  and  plotted.  Since  this  curve 
shows  total  annual  cost  on  a  transformer  and  not  cost  per  1,000 
ft.  of  installation,  a  scale  was  added  on  the  diagonal  at  the  left 
by  use  of  which,  with  a  pair  of  triangles,  the  cost  per  1,000  ft. 
may  be  obtained  by  the  principle  of  similar  triangles. 

8  Yr- 

1,000        YT 

1,000 

.  .YT-  iv  x  -g- 

(YT>  =  total  annual  cost  on  a  transformer) 

(YT  =  annual  cost  of  transformers  per  1,000  ft.  of  installation) 
(see  Fig.  47). 

Hence  by  adding  the  diagonal  scale  at  the  left,  YT  may  be 
obtained  from  YT>  as  follows  by  the  method  of  similar  triangles. 
Draw  a  line  from  the  value  of  YT,  obtained 
on  the  vertical  scale  to  the  value  of  S  used, 
on  the  diagonal  scale.  Draw  a  parallel  line 
through  1,000  ft.  on  the  diagonal  scale  and 
where  it  intersects  the  vertical  scale  will  be 
found  the  desired  value  of  YT. 

Cost   of  Replacing    Transformers. — Two  ^ 

more  items  of  cost  are  of  interest  to  the  QQ> 

designer  and  those  are  arbitrarily  fixed  by  ^ 

local  conditions,  the  cost  of  changing  the 
size  of  transformers  in  the  same  location 
and  the  cost  of  changing  the  location  of  a 
transformer.  These  will  be  practically  con- 
stant for  all  sizes  and  may  be  determined  in  any  case  from 
local  cost  records. 

Application  of  Practical  Curves. — We  are  now  ready  to  furnish 
the  designer  with  the  information  necessary  to  test  the  relative 
economy  of  any  two  alternative  designs.  He  first  determines 
his  wire  size  from  a  study  of  the  theoretical  and  semi-practical 
curves.  Then,  going  to  the  load  curves  he  may  determine  his 
alternatives  in  transformer  size  and  spacing.  Assume  that 
conditions  point  to  the  alternative  of  installing  10-kw.  trans- 
formers at  a  long  spacing,  changing  to  15-kw.  at  a  shorter  spacing 
after  a  certain  period  of  years,  or  of  installing  the  10-kw.  at  the 
shorter  spacing  now  and  merely  changing  sizes  at  that  time. 


164  ECONOMICS  OF  ELECTRICAL  DISTRIBUTION 

From  our  curves  the  exact  cost  per  1,000  ft.  for  each  year  under 
consideration  may  be  obtained  by  using  the  correct  loading  and 
spacing  and,  at  the  proper  time,  adding  the  cost  of  either 
changing  location  or  changing  size.  The  total  of  the  annual 
costs  for  each  design  gives  the  total  cost  over  the  period  under 
consideration  and  a  comparison  of  the  totals  shows  exactly  the 
relative  economy  of  the  designs  over  the  whole  period.  These 
curves  may  be  applied  to  any  such  problem  since  they  are  based 
not  on  the  assumption  of  ideal  conditions  but  cover  any  actual 
condition  which  might  occur  in  practice.  They  can  be  used 
in  cases  where  the  transformer  spacing  cannot  be  uniform  on 
account  of  local  conditions  of  pole  spacing,  secondary  length,  and 
street  and  alley  arrangement,  a  very  usual  case.  When  there  is 
doubt  about  the  wire  size  a  study  of  the  various  possible  combina- 
tions making  use  of  these  curves  will  soon  determine  the  size  for 
greatest  economy.  Similar  curves  can  also  be  developed  to  suit 
other  classes  of  problems  such  as  concentrated  loads,  loads  with 
characteristic  variations  different  from  those  of  the  residence  load 
used  here,  as  in  business  districts,  power  loads,  etc. 

Example  of  Application  of  Practical  Curves. — A  concrete  ex- 
ample of  the  use  of  the  above  curves  may  be  helpful.  Assume 
that  tests  on  a  district  show  a  load  density  of  8  kw.  per  1,000  ft., 
with  No.  4  secondary  wire  already  in  place.  Our  load  curves 
show  for  that  loading  and  size  of  wire,  12.8-kw.  load  at  1,800  ft. 
spacing  to  keep  within  3  per  cent  drop  in  voltage.  We  wish  to 
provide  for  an  increase  in  load  which  we  will  estimate  may  go  to 
15  kw.  per  1,000  ft.  in  6  years.  For  the  present  a  10-kw. 
transformer  spaced  at  1,400  ft.  would  care  for  the  load  while  at 
15  kw.  per  1,000  ft.  there  would  be  required  a  15-kw.  transformer 
at  1,000  ft.  spacing  or  a  25-kw.  at  1,200  ft.  In  order  to  avoid 
too  many  changes  we  may  space  10-kw.  transformers  at  1,000  ft., 
changing  in  3  years  to  15-kw.  or  we  may  put  in  15-kw.  trans- 
formers now  at  1,500  ft.,  changing  the  location  in  2  years  to 
1,000  ft.  Other  alternatives  might  be  considered  but  these  two 
will  serve  as  an  example.  . 

For  the  first  alternative,  assuming  uniform  increase  in  load 
density  of  1%  kw.  per  year. 


SECONDARY  DISTRIBUTION— SINGLE-PHASE  165 

First  year. 

Line  cost  -  LD  =  8  kw.,  S  =  1,000 $16.00  per  1,000  ft.  installation 

Transformer  cost  —  10  kw.  at  8-kw.  load.  ...      32 .  00  per  1 , 000  ft.  installation 


For  year $  48. 00 

Second  year. 

Line  cost  -  LD  =  9%  kw.,  S  =  1,000 16. 30  per  1,000ft. 

Transformer  cost  -  10  kw.  at  9% 32 .  70  per  1 ,000  ft 


For  year $49. 00 

Third  year. 

Line  cost  -  LD  =  \®H  kw.,  S  =  1,000 16.80  per  1,000  ft. 

Transformer  cost  -  10  kw.  at  10^ 33 . 70  per  1,000  ft. 


For  year $50. 50 

Feurth  year. 

Line  cost  -  LD  =  12K  kw.,  S  =  1,000 17.30  per  1,000  ft. 

Transformer  cost  -  15  kw.  at  12^ 42. 90  per  1,000  ft. 

Cost  of  changing  size   (10  kw.  to   15  kw.   on 

same  pole) 7 . 00  per  1,000  ft. 


For  year $67 . 20 

Fifth  year. 

Line  cost  -  LD  =  13%  kw.,  S  =  1,000 17.80  per  1,000  ft. 

Transformer  cost  15  kw.  at  13H 43 . 60  per  1,000  ft. 

For  year $61 . 40 

Sixth  year. 

Line  cost  -  LD  =  15  kw.,  S  =  1,000 18.40  per  1,000  ft. 

Transformer     cost  -  15  kw.  at  15 44 . 40  per  1,000  ft. 


For  year $  62.80 


Total  for  6  years $338. 90    per  1000  ft. 

Second  alternative 
First  year. 

Line  cost  -  LD  =  8  kw.,  S  =  1,500  17.20  per  1,000  ft. 

Transformer  cost  —  15  kw.  at  12-kw.  load. .    35.50  per  1,000  ft. 

For  year $  52.70 

Second  year. 

Line  costLz?  =  9%  kw.,  S  =  1,000 $18.00  per  1,000  ft. 

Transformer  cost  15  kw.  at  15^ 37  .  20  per  1,000  ft. 


For  year $  55.20 

Third  year. 

Line  costLo  =  WH  kw.,  S  =  1,000 16.80  per  1,000  ft. 

Transformer  cost  —  15  kw.  at  10% 42.30  per  1,000  ft. 

Cost  of  changing  location 20.  50  per  1,000  ft. 


For  year $79. 60 

Fourth  year — same  as  first  alternative  (less  charge  for  changing  size)  $60.20  per  1.000  ft. 

Fifth  year — same  as  first  alternative $61 .40  per  1,000  ft. 

Sixth  year — same  as  first  alternative $62 .  80  per  1,000  ft. 


Total  for  6  years $371 . 90  per  1,000  ft. 


166  ECONOMICS  OF  ELECTRICAL  DISTRIBUTION 

A  saving  of  $33  per  1,000  ft.  of  installation,  or  approximately 
10  per  cent  of  the  total  cost  over  a  period  of  6  years  by  the  first 
method  thus  demonstrating  its  economy.  It  is  well  to  note  that 
a  large  part  of  the  difference  in  cost  is  due  to  the  fact  that  in  the 
first  case  the  size  of  transformer  is  changed  while  in  the  other 
the  location  but  not  the  size  is  changed.  If  a  further  refinement 
of  the  comparison  is  desired,  interest  may  be  considered  on  the 
yearly  items  up  to  the  end  of  the  period  under  consideration. 
Usually  such  refinement  is  not  necessary  however. 

Conclusions. — A  study  of  all  these  curves  gives  considerable 
aid  in  determining  certain  standards  of  design  as  well  as  the  final 
particulars  for  any  special  problem.  There  also  may  be  obtained 
a  definite  knowledge  of  the  behavior  of  secondaries  under  various 
conditions  of  loading  and  operation.  It  is  purposed  here  to  take 
up  each  curve  in  detail,  to  bring  out  its  characteristics  and  its 
possible  use. 

Most  Economical  Voltage  Drop. — The  curves  on  voltage  drop 
show  that  the  most  economical  condition  varies  inversely  as  the 
cube  root  of  the  wire  size  also  inversely  as  the  cube  root  of  the 
load  density. 

For  low  load  densities  the  economical  drop  is  high  but  decreases 
rapidly,  while  at  high  loading  the  decrease  is  comparatively  slow. 
It  is  clearly  shown  that  the  most  economical  voltage  drop  may  be 
well  under  that  allowable  for  good  service  for  loads  which  may  be 
often  encountered  in  practice.  Under  the  conditions  and  prices 
assumed  in  the  present  case  the  3  per  cent  limit  seems  to  have 
some  justification  by  economy  for  ordinary  loads. 

Two  conditions  must  be  considered  which  might  affect  these 
curves,  i.e.,  the  price  of  materials  and  cost  of  energy  and  the  fact 
that  here  the  transformer  was  considered  just  sufficient  to  carry 
the  load  while  ordinarily,  when  designing  for  an  increasing  load, 
the  transformers  are  underloaded.  It  is  seen  from  the  equation 
of  the  curves  for  economical  voltage  drop  that  the  cost  of  copper 
does  not  affect  this  discussion.  This  is  due  to  the  fact  that  the 
annual  cost  is  based  on  a  unit  of  1,000  ft.  hence  for  any  given  price 
of  copper  the  annual  cost  per  1,000  ft.  of  three-wire  line  is  a 
constant  no  matter  what  the  load.  The  cost  of  energy  enters  as 
an  inverse  factor  to  the  %  power.  Also  it  is  a  small  element  in 
the  factor  KI,  which  is  also  to  the  %  power  but  in  the  direct  ratio. 
Hence  an  increase  in  the  cost  of  energy  would  increase  both  the 
numerator  and  the  denominator  but  the  latter  slightly  more  than 


SECONDARY  DISTRIBUTION— SINGLE-PHASE  167 

the  numerator,  hence  all  the  curves  would  be  raised  slightly. 
This  effect  would  be  small,  however,  for  ordinary  fluctuations. 
In  the  case  of  an  increase  in  the  transformer  price  there  would  be 
no  change  in  the  curves  providing  the  increase  were  proportional 
to  the  size  since  the  factor  K\  would  not  be  affected  by  such  an 
increase. 

In  ordinary  design  for  an  increasing  load  the  transformer  would 
be  made  larger  than  sufficient  to  carry  the  present  load  to  allow 
for  the  anticipated  increase.  A  study  of  the  curves  for  the  vari- 
ous components  of  the  annual  charge  on  a  transformer  and  the 
equation  resulting  therefrom,  YT'  =  KI  +  KZT,  will  show  that 
if  they  are  developed  with  the  transformer  working  below  its 
rated  loading,  and  if  the  percentage  of  underloading  is  kept  the 
same  for  all  sizes,  the  factor  K\  will  be  very  little  affected,  the 
effect  being  similar  to  an  increase  in  price  proportional  to  size. 
Since  this  is  the  only  part  of  this  equation  that  enters  into  the 
equation  for  most  economical  voltage  drop  it  follows  that  if  a 
design  could  be  limited  to  any  given  percentage  of  underloading 
throughout,  the  curves  would  still  show  the  most  economical 
condition  of  voltage  drop. 

Most  Economical  Transformer  Spacing. — These  curves  for  the 
most  economical  transformer  spacing  (Fig.  38)  are  derived  from 
the  same  general  expression  for  annual  cost  per  1,000  ft.  as  those 
for  economical  voltage  drop.  Hence,  the  same  results  might  be 
expected  from  the  use  of  either  of  these  sets  of  curves  with  the 
exception  that  where  the  most  economical  spacing  would  give  a 
maximum  voltage  drop  of  more  than  3  per  cent  we  have  corrected 
it  for  that  value  making  it  such  as  to  give  3  per  cent. 

These  curves  show  for  very  low  load  densities,  extremely  high 
spacings  which  are  probably  much  greater  than  it  would  be 
practicable  to  use  since  for  such  a  distance  and  such  light  loads 
the  effect  of  the  non-uniform  loading  would  be  considerable.  As 
is  shown  by  the  equation,  the  spacing  for  3  per  cent  drop  varies  as 
the  square  root  of  the  wire  size  while  for  greatest  economy  it 
varies  as  the  cube  root.  It  also  varies  inversely  with  the  load 
density,  to  the  square  root  in  the  first  case,  the  %  power  in  the 
second.  For  ordinary  loadings  encountered  in  practice  and  the 
usual  range  of  wire  sizes  it  is  seen  that  spacing  of  from  1,000 
to  2,000  ft.  is  the  most  economical  and  practicable.  For  the 
higher  loadings  the  most  economical  spacing  decreases  very 
slowly,  remaining  over  500  ft.  up  to  high  values  of  LD. 


168  ECONOMICS  OF  ELECTRICAL  DISTRIBUTION 

Changed  conditions  would  have  a  similar  effect  on  these  curves 
as  on  those  for  economical  voltage  drop,  in  the  range  of  values  for 
which  the  most  economical  voltage  drop  governs  the  spacing. 
That  is,  a  rise  in  the  price  of  energy  would  lower  the  curves 
slightly,  the  prices  of  wire  and  transformers  would  not  have  a 
noticeable  effect.  For  the  condition  of  underloaded  transformers, 
if  the  proportion  of  underloading  were  fixed  there  would  be  slight 
change.  In  practical  designing,  however,  when  considering  the 
amount  of  this  margin  in  transformer  capacity  to  be  used,  it 
might  be  found  relatively  more  economical  to  use  a  transformer 
size  somewhere  near  the  theoretically  most  economical  and  obtain 
the  margin  in  capacity  by  using  a  spacing  less  than  the  most 
economical  spacing.  This  may  have  some  advantage  over  using 
the  most  economical  spacing,  as  shown  by  the  curves,  and  a  larger 
size  of  transformer  than  the  most  economical,  when  the  design 
is  to  cover  several  years  and  the  cost  of  changing  sizes  and  loca- 
tions is  taken  into  account.  Hence  care  must  be  used  in  placing 
too  much  dependence  on  the  strictly  theoretical  values  in 
practical  design.  The  choice  must  be  tested  by  the  actual  year 
to  year  costs  as  shown  by  the  cost  curves. 

Most  Economical  Transformer  Size. — The  curves  for  the  most 
economical  transformer  size  simply  show  the  size  of  transformer 
which  will  carry  the  load  when  the  spacing  is  the  most  economical 
or  just  enough  to  give  3  per  cent  voltage  drop.  They  have 
relatively  less  practical  value  excepting  that  it  is  from  these  and 
the  spacing  curves  combined  that  the  practical  load  curves  are 
obtained. 

Most  Economical  Wire  Size. — The  wire  size  is  the  first  thing  to 
determine  in  a  design  and  must  be  chosen  to  cover  long  periods  of 
increase  in  load  as  replacement  of  secondary  wire  is  very  costly. 
Hence  for  secondaries  a  standard  must  be  chosen  for  installation 
in  new  work  which  will  show  good  economy  through  the  greatest 
range  of  conditions  to  be  encountered.  The  curves  seem  to 
indicate  clearly  that  under  the  conditions  and  prices  assumed 
No.  6  wire  should  be  used  as  a  standard  in  all  new  work,  in  dis- 
tricts where  ordinary  residence  lighting  load  is  expected.  The 
economy  curve  rises  very  rapidly  at  low  densities  up  to  about 
20,000  cir.  mil.  or  nearly  No.  7  at  about  7  kw.  per  1,000  ft. 
From  here  the  rise  is  less  rapid  but  still  considerable  until  it 
crosses  the  value  of  No.  6  wire  at  15-kw.  load  density.  The 
load  density  of  15  is  a  normal  loading.  It  would  not  be  ad  vis- 


SECONDARY  DISTRIBUTION— SINGLE-PHASE  169 

able  to  use  any  size  less  than  a  No.  7  since  the  loadings  at  the 
smaller  values  are  subject  to  such  rapid  increase.  Even  at  No.  7 
the  economical  load  is  fairly  small  (7  kw.  per  1,000  ft.).  On 
the  other  hand,  the  curve  rises  slowly  after  passing  No.  6  and 
only  reaches  No.  5  at  a  loading  of  about  31  kw.  and  No.  4  at 
40  kw.  which  are  high  densities  and  to  be  encountered  only  in 
special  cases.  It  is  interesting  to  note  that  for  all  values  below  a 
No.  6  wire  the  economical  size  is  governed  by  3  per  cent  voltage 
drop  while  above  that  the  most  economical  drop  governs,  the 
curves  crossing  at  19-kw.  load  density. 

Since  the  curves  were  figured  at  a  low  copper  price,  in  case  of  an 
increase  in  price,  the  curves  would  be  lowered,  i.e.,  a  smaller  wire 
size  would  be  indicated  for  any  particular  loading.  An  increase  in 
energy  cost  would  slightly  raise  the  curve,  an  increase  in  trans- 
former price  if  proportional  to  size  would  not  affect  the  discussion. 
Since  the  curves  were  figured  on  the  assumption  that  the  trans- 
former spacing  was  the  most  economical  and  the  size  just  equal  to 
the  load,  a  change  in  these  conditions  might  affect  the  most 
economical  wire  size  somewhat.  A  fixed  proportion  of  under- 
loading as  above  shown  would  have  little  effect  but  if  different 
conditions  of  spacing  were  assumed,  the  design  should  be  tested 
by  use  of  the  cost  curves  for  various  sizes  of  wire. 

Semi-practical  Curves. — The  curves  which  we  call  semi- 
practical  show  a  little  more  concretely  the  relative  economy  of 
installations  with  the  various  sizes  of  wire,  in  dollars  per  year 
per  1,000  ft.  They  show  the  actual  annual  cost  for  different 
types.  The  excessive  cost  of  No.  2  wire  for  ordinary  loads  is 
clearly  demonstrated  being  from  $3.50  to  $6.00  per  year  more 
than  No.  4  for  loadings  up  to  15  kw.  per  1,000  ft. 

When  we  go  from  the  ideal  size  of  transformer  to  practical  stock 
sizes,  still  assuming  the  best  spacings  to  be  used,  there  are  some 
conditions  in  which  the  relative  wire  economy  is  somewhat 
different.  These  curves  also  give  an  indication  of  transformer 
economy.  It  seems  to  be  quite  clearly  shown  that,  under  the 
assumed  conditions,  the  use  of  large  transformers  such  as  25  kw. 
is  not  justified  except  with  very  heavy  loading,  even  considering 
the  possible  reduction  in  the  number  of  transformers  and  hence 
in  the  core  loss.  The  increase  in  the  investment  cost  more  than 
equalizes  such  saving. 

Practical  Curves. — The  use  of  the  cost  curves  in  designing  has 
already  been  explained.  It  may  now  be  readily  seen  how  a  study 


170  ECONOMICS  OF  ELECTRICAL  DISTRIBUTION 

of  the  theoretical  and  semi-practical  curves  applied  to  any 
problem  will  give  a  basis  upon  which  to  formulate  a  design  which 
can  then  be  tested  for  actual  economy  by  application  of  the  exact 
costs  to  be  expected.  We  can  determine  from  this,  in  case  of  a 
new  line,  the  size  of  wire  and  then  the  spacing  and  size  of  trans- 
formers which  will  care  for  several  years  of  increase.  The  exact 
number  of  years  will  be  determined  by  the  rate  of  increase 
together  with  the  economy  of  the  design,  including  cost  of 
changing  sizes  and  locations.  Or,  in  case  of  remodeling  an  old 
district,  we  start  with  a  given  size  of  wire  which  although  perhaps 
not  the  most  economical,  will  not  justify  the  cost  of  change.  We 
can  then  choose  and  space  our  transformers  most  economically 
with  regard  to  that  size  of  wire.  In  a  special  case  where  no 
increase  in  load  is  expected  the  theoretical  curves  will  give 
exactly  the  design  to  use.  In  other  cases  where  the  loading, 
voltage,  etc.,  are  somewhat  different,  by  proper  substitution  in 
the  theoretical  formulae,  curves  could  be  plotted  which  would 
apply  to  that  particular  condition. 

General. — The  curves  given  here  should  not  be  accepted  for 
general  application  to  design  problems.  The  costs  and  condi- 
tions of  loading  used  were  of  local  derivation  and  apply  only  to 
the  organization  and  the  time  for  which  they  were  obtained. 
Similar  curves  should  be  developed  for  the  study  of  conditions  in 
any  other  locality  and  they  should  be  revised  from  time  to  time 
to  meet  changing  conditions.  These  examples  are  given  here 
merely  to  indicate  the  characteristics  of  such  curves. 

It  is  evident  that  no  very  simple  means  of  correctly  designing 
a  distribution  system  in  regard  to  transformers  and  secondary  wire 
can  be  made  available  due  to  the  many  varying  conditions  encoun- 
tered and  the  large  number  of  factors  to  be  taken  into  account. 

The  elements  of  good  judgment  and  experience  are  as  necessary 
in  the  solution  of  these  problems  as  in  any  other  problem  of 
engineering.  The  object  of  this  study  has  been  to  analyze  and 
evaluate  the  factors  of  the  design  of  a  single-phase  secondary 
system  of  the  type  considered,  that  lend  themselves  to  such 
definite  analysis  and  to  present  the  results  as  aids  in  the  appli- 
cation of  good  judgment  and  experience  to  the  best  possible 
solution  of  the  problem.  This  problem  has  been  dwelt  on  in  some 
detail  since  it  is  thought  that  the  methods  used  and  the  principles 
brought  out  are  typical  for  a  large  number  of  such  types  of 
problems. 


SECONDARY  DISTRIBUTION— SINGLE-PHASE  171 

Secondaries  for  Scattered  Load. — A  very  common  question 
arising  in  rural  districts  where  the  load  is  scattered,  is  that  of 
how  far  it  is  economical  to  extend  secondary  from  a  present 
transformer  location  to  reach  a  new  customer  rather  than  to  hang 
a  new  transformer.  It  is  thought  that  this  problem  is  of  suffi- 
cient interest  to  warrant  a  brief  mention  here. 

The  comparison  should  be  made,  of  course,  on  the  basis  of 
annual  costs. 

The  annual  cost  on  the  installation  in  case  the  secondary  is 
extended  is 

Y8  =  annual  charges  on  investment  on  line  of  length 

D  -  Ds 

-f-  cost  of  PR  loss  on  extended  secondary 
-f-  cost  of  increase  in  I2R  loss  in  transformer 
-+-  cost  of  increased  copper  loss  in  transformer.    (72) 
Where  D  =  distance  from  present  transformer  to  new  load, 

Ds  =  distance  from  present  transformer  end  of  present 

secondary, 

(It  is  assumed  that  the  present  transformer  is  large  enough  to 
carry  the  increased  load.) 

The  annual  cost  in  case  the  primary  is  extended  and  a  new 
transformer  used,  is 

Yp  =  annual  charge  on  investment  on  line  of  length 

D  -  Dp 

+  I2R  loss  in  total  length  of  primary  D 
-f-  total  annual  charges  on  new  transformer  including 
fixed  charges  and  energy  losses.  (73) 

Where  Dp  =  distance  from  present  transformer  to  end  of  primary. 
In  the  problem  considered,  it  is  assumed  that  the  cost  of  right- 
of-way,  poles,  crossarms,  etc.  would  be  the  same  in  either  case. 
If  the  expressions  for  F8  and  Yp  are  put  in  the  form  of  equa- 
tions in  terms  of  the  load,  voltage,  wire  size,  cost  of  energy,  etc. 
and  Fs  equated  to  Yp,  a  solution  may  be  obtained  for  D,  the 
distance   at   which  economy   changes  from   a  secondary  to   a 
primary  extension. 

The  following  expression  was  obtained  using  constants  apply- 
ing to  a  particular  system. 

(TF2\   /  1          1   \  W 

•457  —-)  (±  --J-)  =  -26(Z>.  -  Dp)  (.175w>  +  .0125)  -  .021l|-2 

(21.6  ^^  +  (W+  2TFi)  Rn  -  WRn)  +  K 


172  ECONOMICS  OF  ELECTRICAL  DISTRIBUTION 

Where  W  =  new  load  in  watts, 

E  =  the  secondary  voltage, 
EI  =  the  primary  voltage, 

w  =  weight  per  foot  of  conductor  (primary  and  sec- 
ondary  assumed   same   size   for   small  load), 
Wi  =  present  load  on  present  transformer, 
Rti  =  equivalent  resistance  of  present  transformer, 
Rtz  =  equivalent  resistance  of  new  transformer, 
K  =  fixed  charges  and  annual  cost  loss  cost  on  trans- 
former, 

If  both  primary  and  secondary  were  two  No.  6  wire  and  EI  = 
4,600,  E  =  112  —  the  equation  becomes 

L3821) 


W         2W,\  W 

n  ~          " 


oo 

In  the  limiting  case,  where  Dp  =  D  and  Ds  =  0  the  cost  of 
primary  extension  would  be  a  minimum  and  of  secondary  exten- 
sion a  maximum.  If  TI  —  T2  =  2  kva.  and  TI  is  fully  loaded 


18,550-  4,180 

(74) 


This  indicates  the  distance  to  which  it  is  economical  to  carry 
any  load,  TF,  on  secondary. 

Since,  for  3  per  cent  voltage  drop  with  the  above  conditions 

D=       417 

TF/1,000 
Solving  simultaneously 


1,000 

This  indicates  that,  for  any  load  over  200  watts,  economy 
need  not  be  considered  if  the  maximum  allowable  voltage  drop 
is  fixed  at  3  per  cent,  since  it  is  economical  to  carry  such  a  load  on 
secondary  to  any  distance  at  which  the  voltage  drop  is  3  per  cent 
or  less. 

For  loads  less  than  200  watts,  Eq.  74  may  be  plotted  in  a  curve 
if  desired. 

The  above  solution  has  been  given  very  briefly  and  all  but  the 
chief  steps  omitted.  It  indicates  the  method  which  can  be 


SECONDARY  DISTRIBUTION— SINGLE-PHASE  173 

followed  in  studying  a  number  of  similar  problems.  The  rule 
established  in  the  particular  case  shown  has  been  found  very 
useful  in  laying  out  rural  extensions. 

Space  will  not  permit  the  elaboration  of  more  problems  of 
single  phase  secondaries.  Among  the  others  often  encountered 
are  the  following :  economy  of  replacing  a  small  secondary  with  a 
larger  size  instead  of  hanging  an  additional  transformer  (see 
Chap.  X  on  " Reconstruction  Problems");  replacing  wire  larger 
than  the  economical  size  with  a  smaller  size;  the  economical  size 
of  secondaries  and  arrangement  of  transformers  for  electric 
range  loads;  economy  of  leaving  dead  wire  in  place  if  it  is  to  be 
utilized  later;  and  many  others. 


CHAPTER  XIV 
POWER  SECONDARIES 

POWER  SECONDARY  vs.  SEPARATE  TRANSFORMERS — ECONOMICAL 
SIZE    OF    30    SECONDARIES 

The  power  secondaries  on  any  system  are,  as  a  rule,  by  no 
means  as  extensive  as  the  lighting  secondaries.  In  districts  where 
the  power  load  is  very  heavy,  it  is  very  often  advisable  to  supply 
each  customer  from  separate  transformers.  In  districts  where 
the  power  load  is  scattered,  the  distances  are  usually  too  great  to 
carry  more  than  a  very  few  customers  on  one  secondary.  There 
are,  of  course,  many  cases  where  several  small  or  medium  sized 
loads  are  grouped  in  a  relatively  small  area,  such  as  a  number  of 
small  factories  or  a  block  of  stores.  For  such  conditions,  it  is 
usually  practicable  to  use  a  few  large  transformer  installations  with 
power  secondaries.  While  the  proportion  of  the  total  system 
investment  represented  in  power  secondaries  in  relatively  small, 
nevertheless  the  amount  of  load  handled  on  any  one  installa- 
tion is  usually  large  compared  with  that  on  a  lighting  secondary. 
Hence,  although  a  consideration  of  their  economy,  as  a  general 
proposition  may  not  seem  so  important,  in  individual  cases  it  may 
be  quite  profitable. 

It  very  rarely  happens  that  the  load  on  a  power  secondary  is 
so  arranged  that  it  may  be  considered  as  a  distributed  load, 
either  uniformly  or  in  accordance  with  any  other  definite  law. 
Neither  does  it  often  occur  that  power  secondaries  can  be  made 
continuous  and  the  transformers  spaced  as  desired.  Hence,  the 
problem  is  usually  one  of  concentrated  loads  of  a  given  size  to  be 
transmitted  a  definite  distance.  There  must  be  a  study  of  each 
particular  case  rather  than  of  the  type  of  installation  in  general. 

Two  kinds  of  problems  are  quite  commonly  met  with  in  connec- 
tion with  power  secondaries.  It  must  be  decided,  for  any  case  of 
a  small  or  medium  sized  load,  whether  it  is  preferable  to  carry  it 
on  a  separate  transformer  or  tap  it  to  a  power  secondary,  provid- 
ing one  is  available  or  can  be  installed.  If  it  is  to  be  thus  handled, 
the  most  economical  size  of  conductor  should  then  be  determined. 

Regulation  and  Continuity  of  Service  Important. — The  deciding 
factor  is  quite  likely  to  be  some  other  consideration  than  that  of 
economy  only.  It  must  first  be  determined  whether  the  load  can 

174 


POWER  SECONDARIES  175 

be  carried  from  the  present  installation,  with  a  reasonable  sized 
secondary,  without  exceeding  the  allowable  voltage  drop.  Often- 
times, a  present  transformer  installation  can  be  moved  to  a  new 
location  and  the  secondaries  rearranged  to  accommodate  such 
additional  loads.  Even  though  it  is  found  easily  possible  to  carry 
the  load  in  this  way  there  are  some  cases  where  the  importance  of 
continuity  of  service  may  indicate  that  a  separate  installation  is 
advisable.  Any  trouble  on  one  customer's  service  or  in  the 
transformer  would  disable  the  services  of  all  other  customers. 
Hence,  any  customer  whose  service  is  especially  subject  to  inter- 
ruptions, or  any  one  whom  an  interruption  would  seriously 
discommode  should  be  given  separate  transformer  installations. 
For  the  same  reason,  it  is  well  not  to  concentrate  too  many  power 
services  on  one  secondary,  especially  services  to  manufacturing 
plants. 

Economy  of  Secondary  Instead  of  Separate  Transformers. — 
If  it  has  been  decided  that  the  load  in  question  can  and  should 
be  carried  on  secondary,  it  still  remains  to  determine  whether 
such  an  installation  is  economical.  If  the  annual  charges 
including  cost  of  energy  loss  on  the  secondary,  necessary  to  reach 
the  customer,  is  greater  than  the  annual  charges  on  a  separate 
transformer  installation,  the  latter  should  be  used.  If  the  whole 
installation  is  to  be  rearranged,  the  annual  charges  on  the  whole 
cost  of  making  the  change  must  be  included.  For  small  loads  or 
loads  near  a  present  transformer,  a  secondary  installation  is,  in 
most  cases,  more  economical.  The  cases  to  be  questioned  are 
those  where  heavy  secondaries  of  considerable  length  are 
necessary. 

For  example,  suppose  a  new  load  of  25  hp.  is  to  be  served,  which 
is  800  ft.  from  a  present  installation.  In  order  to  get  proper 
regulation,  it  will  be  necessary  to  string  No.  0  secondaries  the 
whole  distance.  The  transformer  which  at  present  is  a  75  kva., 
well  loaded,  must  be  changed  to  a  100  kva.  It  will  be  assumed 
that  the  necessary  poles  will  be  the  same  for  either  a  secondary 
or  a  primary  extension  to  the  customer.  To  determine  the  most 
economical  installation,  we  must  compare  the  annual  costs  of 
the  two  alternatives.  These  are  made  up  as  follows: 
Secondary  Installation. — 

1.  The  annual  charges  on  800  ft.  of  three  No.  0  secondary, 
including  cost  of  conductor,  crossarms,  pins,  insulators,  and  labor 
cost  for  installing. 


176  ECONOMICS  OF  ELECTRICAL  DISTRIBUTION 

2.  Annual  cost  of  energy  losses  over  the  new  secondary. 

3.  Annual  charges  on  cost  to  change  transformers. 

4.  Annual  charges  on  increase  in  transformer  investment  from 
a  75  to  a  100  kva. 

5.  Annual  cost  of  increase  in  transformer  energy  losses. 
Primary    Installation. — 

1.  Annual   cost   on   800  ft.   of   primary,   unless   the   present 
primary  extends  past  the  new  load. 

2.  Energy  losses  on  the  primary. 

3.  Annual  cost  on  a  new  transformer  installation  of  25  kva., 
including  losses. 

Some  such  figures  as  the  following  may  be  obtained: 

Secondary  installation  Primary  installation 

No.  1 $  39.00         No.  1 $  15.00 

No.  2 32.00         No.  2 •        2.50 

No.  3 2.00         No.  3 118.00 

No.  4 35.00 

No.  5..                                        14.00  $135.50 


Total $122.00 

The  secondary  installation  will  have  advantage  of  $13.50  per 
year  which  at  15  per  cent  represents  a  capitalization  of  $90.00 

If  the  primary  is  already  in  place,  the  advantage  would  be 
slightly  with  the  separate  installation,  although  the  difference 
in  cost  is  small. 

Economical  Size  of  Secondaries. — When  it  has  been  decided 
upon  that  a  load  is  to  be  carried  on  secondary,  either  after  some 
such  consideration  as  the  above,  or  when,  with  a  separate  installa- 
tion, in  order  to  locate  the  transformer  conveniently,  it  is  necessary 
to  string  a  few  spans  of  secondary  from  the  transformer  to  the 
service,  there  still  remains  the  question  of  the  most  economical 
size  of  conductor  to  be  used.  Let  us  assume  for  this  example 
small  30  power  installations  for  which  the  load  is  considered 
continuous  during  the  time  of  operation. 
Equation  of  Annual  Cost — 

Annual  cost  on  30  secondary  =  r  gr(cost  of  wire  +  cost  of  stringing)  ~| 

per  100ft.     +g(costof  poles,  fixtures  and  guys)    (75) 
L-f-  cost  of  energy  loss. 

Where  g  —  per  cent  interest,  taxes  and  depreciation 
=  13  per  cent  for  wire. 

Assume  poles,  fixtures  and  guys  the  same  for  all  cases  and 
neglect  cost  in  making  comparison.  No  doubt  with  the  heavier 


POWER  SECONDARIES  177 

sizes  of  wire,  additional  cost  will  be  found  necessary  for  heavier 
cross  arms,  insulators,  etc.  and  additional  guying.  The  study 
could  be  made  so  as  to  include  these,  as  was  done  in  the  case  of 
primary  lines  in  Chap.  XI,  but  in  this  case  this  factor  will  be  left 
out  of  the  computations.  In  using  the  resulting  curves  it  can  be 
kept  in  mind  and  will  have  the  effect  of  increasing  slightly  the 
economical  load  on  those  sizes. 

Annual  charge  for  energy  loss  (30)  = 

PR  X  365  X  t  X    ~~    =  .365/2/ttC. 


kw2 
Annual  charge  (energy)  =  365,000  Dr  -™ ^  tCe 

In  the  above  t  =  equivalent  hours  per  day, 

If  tw  =  equivalent  hours  per  week  which  is  assumed  in  this  case 
to  be  hours  per  week  which  motor  runs  and  power  is  given  in 
horsepower. 

E  =  220  volts;  cos  0  =  .80 

fjp*  x  (.746)2 
Annual  charge  =  52,000  Dr ™ '-^ —  twCe 

EJ     COS     (/ 

=  mZDrtwCeHP* 

D  =  distance  one  way  in  hundreds  of  feet. 
r  =  resistance  per  wire  per  100  ft. 
Annual  charge  per  100  ft.  (energy)  =  .933r^Ce#P2 
The  cost  of  the  conductor  per  100  ft.  was  determined  from  the 
number  of  pounds  per  100  ft.  for  different  sizes,  the  cost  of  ties, 
and  the  proportional  charge  for  freight  and  injury  to  returned 
reels.     This  was  put  in  the  form  KmCcu  +  Kn,  so  that  the  effect 
of  a  change  in  price  of  copper  could  be  studied.     The  labor  cost 
was  determined  for  average  conditions  and  reduced  to  a  cost  per 
100  ft.  for  each  size.     The  proper  overhead  loading  percentage  was 
added  to  both  material  and  labor.     Combining  these  two  charges, 
the  expressions  for  annual  charges  on  the  conductor  in  place  for 
each  size  were  determined  as  shown  in  Table  17,  p.  179,  column  2. 
Column  3  of  the  same  table  gives  the  resistance  per  100  ft.  of 
the  various  sizes  of  conductors  and  column  4  the  corresponding 
expression  for  annual  charge  for  energy  loss,  found  by  inserting 
the  proper  value  of  r  in  equation  above.     The  sum  of  columns  2 
and  4  would  give  the  total  annual  charges  per  100  ft.  for  each 
size  of  conductor. 

12 


178 


ECONOMICS  OF  ELECTRICAL  DISTRIBUTION 


Equations  of  Equal  Cost. — If  the  expression  for  the  total 
charges  on  one  size  of  wire  is  equated  to  that  for  any  other  size, 
the-  resulting  equation  would  represent  the  conditions  for  which 
there  is  no  economical  advantage  of  one  size  over  the  other. 
This  is  done  for  each  adjacent  size  in  the  table  and  the  resulting 
expressions  are  given  in  column  5. 

If  we  consider  two  prices  of  copper,  the  results  for  any  inter- 
mediate price  can  be  interpolated.  In  the  table  below  the 
expressions  for  equal  annual  cost  are  given  with  the  copper  price 
substituted,  using  prices  of  20  cts.  and  30  cts.  per  pound. 

TABLE  18 


Size  of 
Wire 

Size  of 
Wire 

If  Ccl 

,  =  .30 

If  Cc 

t  =  -20 

6         to             4 

twCeHP* 

=      55.99 

twCeHP* 

=      39.04 

4         to             2 

twCeHP2 

=     165.77 

twCeHP2 

=     113.47 

2         to             0 

twCeHP* 

=    465.00 

twCeHP* 

=    317.20 

0         to           00 

twCeHP* 

=    788.50 

twCeHP2 

=    547.50 

00         to         000 

twCeHP* 

=  1276.80 

twCeHP* 

=    889.80 

000         to       0000 

twCeHP* 

=  1784.40 

twCeHP* 

=  1214.40 

Value  of  t«,Ce. — In  previous  chapters  the  variation  of  the  cost 
of  energy  with  the  load  factor  and  hence  with  the  equivalent  hours 
has  been  explained.  In  "Appendix  A"  the  method  of  approxi- 
mating the  value  of  twCe  is  carried  out.  In  this  case,  the  loads  are 
considered  small,  consisting  of  not  more  than  two  or  three  motors, 
which  would  give  a  flat  load  curve.  The  value  of  equivalent 
hours  per  week,  tw,  would  then  be  practically  equal  to  the  aver- 
age number  of  hours  per  week  which  the  motors  are  run.  In. case 
of  loads  of  a  different  character,  a  more  accurate  determination  of 
tw  would  be  necessary.  The  values  of  tw  for  any  load  factor,  and 
the  corresponding  values  of  twCe  used  are  given  below. 

TABLE  19 

tw  LOAD  FACTOR  t^Ce 

.0  .0                                         .0 

16.8  .10                                       .467 

33.6  .20  .619 

50.4  .30  .756 

67.2  .40  .894 

84.0  .50  1.015 

100.8  .60  1.139 

117.6  .70  1.258 

134.4  .80  1.358 

151.2  .90  1.450 

168.0  1.00  1.543 


POWER  SECONDARIES 


179 


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180 


ECONOMICS  OF  ELECTRICAL  DISTRIBUTION 


The  curve,  Fig.  48  is  plotted  from  the  above  figures,  giving 
the  value  of  twCe  for  any  value  of  tw. 

Curves  for  Economical  Conductor  Size. — The  two  sets  of 
curves  given  in  Fig.  49  and  Fig.  50,  can  now  be  plotted  from  the 
equations  in  Table  18,  using  as  coordinates  the  average  number  of 
hours  of  operation  per  week  and  the  load  in  horsepower  (the  load 
is  given  in  horsepower  for  convenience  in  use  with  loads  consist- 
ing of  one  or  two  motors  which  are  so  rated).  The  curves  are 
similar  to  those  given  for  " Power  Lines"  in  Chap.  XI.  The 
conductor  sizes  given  pertain  to  the  areas  between  the  curves, 


^ 

^ 

' 

^--* 

.  ' 

~^"^ 

. 

^ 

•^"^ 

^ 

^ 

^ 

^ 

Value 
value 

i  Of 

soff 

^Q. 
»(ho 

for  & 
urs  p 

ifftn 
°rwe 

/ 

^ 

*) 

/ 

/ 

Hours  per  Week 

FIG.  48. — Values  of  twCe  for  different  values  of  tw   (hours  per   week). 

since  the  curves  themselves  are  the  loci  of  points  of  equal  cost. 
For  example  (using  20-ct.  copper)  three  No.  2  secondary,  is  more 
economical  than  three  No.  4  for  a  20-hp.  load,  if  it  operates  more 
than  6J^  hr.  per  week.  It  is  less  economical  than  three  No.  0 
if  the  20-hp.  motor  operates  more  than  54  hr.  per  week.  Simi- 
larly for  a  load  operating  4  hr.  per  week,  three  No.  0  is  the  most 
economical  secondary  for  loads  between  21J^  hp.  and  28 J^  hp. 
If  higher-priced  copper  is  being  used,  the  corresponding  loads  and 
hours  of  operation  will  be  larger,  as  shown  by  Fig.  50.  The 
distance  which  the  point  representing  a  given  load  lies  from  either 
boundary  curve  is  an  indication  of  the  amount  of  economical 
advantage  of  the  size  of  wire  shown,  over  the  next  adjacent  size. 
Naturally,  these  curves  apply  only  to  concentrated  loads  of 
the  character  assumed.  Similar  curves  may  be  developed  for 
any  type  of  load  desired  by  the  inclusion  of  the  proper  factors  in 
making  up  the  equation.  For  loads  which  are  not  concentrated, 
the  secondary  may  be  studied  by  dividing  into  sections.  Curves 
for  large  power  loads  can  be  similarly  developed.  In  that  case, 
however,  it  is  probably  better  to  designate  the  load  in  kilowatts 


POWER  SECONDARIES 


181 


120 


110 


S'zes  of  wire  shown  belong  to  AREAS  BETWEEN  CURVES. 
Points  on  curves  represent  conditions  for  which  the  annual 
•cost Is  the  same  for  either  adjacent  wim&ize. 


ASSUMPTIONS: -Load 'consists  of  one  motor  ore 
~no  variation  from  full  load  while 


alent 


•yrequ/yalenf 
'leitison.  "T 


Cost  of  construction  other  than  wire  stringing 
-same,  for  all  sizes. 


'0        10      20      30      40      50      GO      10      80      90.     100      110      120      130      140     150      160     110 

Hours  per  Week 

FIG.  49. — Curves    showing    most    economical    size    of    wire    for    three-phase 

secondaries. 


I     I     !     I     I     1     I 

Sizespfwire  shown  belong fo AREAS  BETWEEN  CURVES. 


Points  ori  curves  represent  conditions  for  which  the  annual  cost 
s  the  same  for  either  adjacent  wire  size. 


ASSUMPTIONS'.  -Load  consists  of  one  motor  or  equivalent  no.  variatior 
from  full  load  while  it  is  on. 
Cos  tof  construction  other  than  wire  •stringing 
same  for  all  sizes. 
Copper  at  3  0  cents  per  Ib 


0       10      20      30      40       50      60      10      80      90      100     HO      120     130      140      150     ISO     HO     180 

Hours  per  Week 

FIG.  50. — Curves    showing    most    economical    size    of    wire    for    three-phase 

secondaries. 


182  ECONOMICS  OF  ELECTRICAL  DISTRIBUTION 

and  use  values  of  t  =  equivalent  hours  per  day,  as  was  done  in  the 
case  of  power  primaries,  Chap.  XI. 

Problems  in  Power  Secondaries  Comparatively  Simple. — The 
problems  relating  to  power  secondaries  are  comparatively  simple. 
With  good  cost  data  on  the  installation  of  such  secondary  and 
transformers,  load  curves  for  power  secondaries  as  shown  in 
Chap.  VII,  and  curves  for  economical  conductor  size,  nearly  all 
such  problems  as  are  discussed  in  this  chapter  may  be  readily 
solved.  Power  secondaries  may  be,  of  course,  a  relatively  less 
important  part  of  the  system,  than  transmission  lines,  primaries 
or  lighting  secondaries.  Notwithstanding  this  fact,  however, 
the  study  of  their  economy  should  not  be  overlooked. 


CHAPTER  XV 
UNDERGROUND  LINES 

VOLTAGE — CABLE  SIZE — ROUTE — NUMBER     OF     DUCTS    IN    A 
DUCT  LINE — ARRANGEMENT   OF   DUCTS   AND   CABLES 

The  major  part  of  this  book  has  been  devoted  to  the  problems 
connected  with  overhead  lines.  This  was  done  not  because  the 
economic  study  of  underground  lines  is  any  less  important  than 
that  of  overhead.  The  opportunities  for  effecting  economies  are 
just  as  great,  or  perhaps  greater,  on  underground  lines  on  account 
of  the  greater  cost  of  construction.  It  is  a  fact,  however,  that  on 
most  of  the  central-station  systems  in  this  country,  the  under- 
ground lines  are  of  small  extent  as  compared  with  the  overhead. 
Overhead  work  is  preferred  where  possible,  underground  being 
used  chiefly  in  congested  down-town  areas  and  for  transmission 
lines  where  high  voltage  is  not  permitted  by  the  municipality,  or 
is  considered  unsafe.  In  some  other  countries,  the  case  is  quite 
different,  underground  work  predominating.  In  any  case,  the 
principles  employed  in  solving  the  problems  of  underground  lines 
are  the  same  as  those  used  for  overhead  which  have  been  ex- 
plained and  exemplified.  Only  the  conditions  of  the  individual 
problems  are  different.  Some  of  the  affecting  factors  and  special 
problems  relating  to  underground  lines  will  be  taken  up  in  this 
chapter. 

The  necessity  rarely  arises  to  make  a  choice,  on  the  basis  of 
economy,  between  an  underground  installation  and  an  overhead. 
Overhead,  as  a  rule,  costs  only  a  fraction  of  the  amount  necessary 
to  install  underground  for  the  same  load.  An  overhead  trans- 
mission line,  for  example,  can  be  built  for  between  $3,000  and 
$4,000  per  mile  to  carry  the  load  which  would  require  an  under- 
ground installation  costing  $15,000  to  $18,000  per  mile.  Over- 
head also  has  the  advantage  of  greater  flexibility  when  changes 
are  necessary  to  accommodate  increases  in  load.  The  choice  of 
underground  is  usually  based  on  considerations  of  necessity  (in 
congested  areas),  safety,  sightliness,  etc. 

The  nature  of  the  construction  on  underground  lines  makes 
the  necessity  for  standardization  and  for  the  best  possible  work- 
manship very  evident.  The  comparative  inaccessibility  and 

183 


184  ECONOMICS  OF  ELECTRICAL  DISTRIBUTION 

the  high  cost  of  installation  make  repair  work  very  expensive. 
True  economy  lies  in  using  all  means  possible  to  reduce  such 
repair  work  to  a  minimum. 

Problems  Similar  to  those  of  Overhead  Lines. — A  great  many 
of  the  problems  of  underground  lines  are  very  similar  to  those 
already  discussed  for  overhead.  There  is,  for  example,  the  same 
general  classification  into  transmission  lines,  primaries  and 
secondaries,  with  the  special  questions  arising  with  each.  The 
economy  of  any  installation  should  be  similarly  studied  with 
respect  to  voltage,  voltage  drop,  conductor  size,  transformer  size 
and  location,  most  economical  route,  etc.  The  solution  of  these 
problems  with  respect  to  underground  lines  is  limited  by  their 
character.  Inflexibility  and  high  cost  of  construction  make  it 
necessary  that  original  installations  be  designed  with  sufficient 
thought  toward  probable  future  conditions,  even  at  the  expense 
of  apparent  present  economy  in  some  cases.  There  are  some 
other  problems  concerning  the  construction  itself,  which  apply 
only  to  underground  lines,  such  as  the  number  and  arrangement 
of  ducts  in  a  run,  and  of  cable  in  the  ducts,  etc.  These  will  be 
taken  up  after  some  of  the  questions  of  voltage,  conductor  size, 
route,  etc.  have  been  briefly  considered. 

Voltage. — The  determination  of  the  most  economical  voltage 
is  rarely  dependent  on  the  economics  of  the  underground  system 
alone.  In  transmission,  the  voltage  is  limited  by  the  type  of 
cables  available.  At  present,  about  33,000  volts  is  the  practi- 
cable limit,  although  higher  voltages  are  being  considered  and  will 
probably  be  used  in  the  near  future.  For  power  lines,  one  or  two 
standard  voltages  are  usually  chosen  for  the  whole  system. 
While  the  economics  of  the  underground  lines  should  be  con- 
sidered in  this  choice,  it  is  by  no  means  the  only  factor.  A 
study  of  the  system  as  a  whole  is  necessary. 

Conductor  Size. — A  new  element  is  introduced  in  the  study  of 
the  most  economical  cable  to  carry  any  load  or  the  most  econo- 
mical load  for  any  cable.  The  limiting  factor  in  this  case  is 
usually  the  current  carrying  capacity  of  the  cable.  This  is 
governed  by  the  heating  of  the  cable.  It  depends  not  only  on  the 
size  of  conductor  and  on  the  insulation  but  on  the  location  of 
the  cable  in  the  duct  run,  the  number  of  ducts,  the  condition 
of  the  surrounding  soil  and  the  shape  of  the  curve  of  the  load 
on  the  cable  itself  and  on  the  other  cables  in  adjacent  ducts. 

The  question  of  the  heating  of  cables  has  been  attracting  a 


UNDERGROUND  LINES 


185 


great  deal  of  attention  recently  and  a  number  of  articles  on  this 
subject  have  appeared  in  the  Journal  of  the  A.  I.  E.  E.  and  other 
publications.  The  study  is  still  in  the  making  and,  while  much 
valuable  data  has  been  collected,  there  is  still  much  to  be  done. 
At  present  there  is  no  generally  accepted  standard  on  such 
matters.  It  is  safe  to  say  that  in  practically  all  cases  of  ordinary 
loads,  the  apparent  economical  load  for  a  cable,  basing  the  figures 
on  the  normal  life  of  that  cable,  will  be  considerably  more  than 
the  safe  current  carrying  capacity,  if  the  cable  is  to  fulfill  its 


8.00 


TOO 


6.00 


500 


~Z    4.00 


3.00 


150  200 

load  in  Amperes 


FIG.  51. — Annual  cost  per  1,000  feet  per  ampere  transmitted  No.  00  underground 
cable  23,000  volts. 


normal  life.  For  example,  Fig.  51  shows  the  annual  cost  of 
transmitting  loads  over  No.  00  underground  cable  at  23,000 
volts  for  various  values  of  equivalent  hours.  In  making  the 
computations,  depreciation  was  figured  on  the  basis  of  a  20-year 
life  for  the  cable.  It  is  seen  that  the  most  economical  loadings 
on  that  basis  would  be  between  200  and  300  amp.,  depending  on 
the  load  factor  (or  the  equivalent  hours).  Figure  52  gives  the 
average  allowable  load  for  such  a  cable  in  various  combinations 
with  similar  cables,  based  on  the  heating  of  the  cable.  It  shows 
that  the  safe  loading  in  duct  runs  of  the  usual  size  is  in  the 
neighborhood  of  only  100  amp.,  or  about  50  per  cent  of  the  most 
economical  load,  as  shown  by  Fig.  51.  It  might  be  thought 


186 


ECONOMICS  OF  ELECTRICAL  DISTRIBUTION 


possible  to  run  the  cable  at  a  higher  average  load  than  that  given 
as  the  safe  load,  letting  the  increased  depreciation  due  to  the 
shortening  of  the  life  of  the  cable  be  balanced  against  the  in- 
creased economy  of  operation.  Not  enough  data  is  available  on 
the  extent  to  which  a  cable  is  damaged  by  an  overload  to  warrant 
any  figures  on  this  subject.  It  is  quite  probable  that  the  damage 
done  would  be  greater  than  the  economy  effected  in  most  cases. 
In  the  example  represented  by  Fig.  51,  if  the  load  were  such  as 
to  reduce  the  life  of  the  cable  to  10  years,  the  apparent  most 
economical  load  at  tCe  =  .10  would  be  about  225  amp.  How- 
ever, if  a  load  of  140  amp.  or  greater  could  be  carried  without 
reducing  the  life  of  the  cable  to  less  than  10  years,  the  annual 


5        6        7        8        9        IO       II        12 
No. of  Ducts  and  Cables  per  Duct  Line 


FIG.  52. — Curves  showing  current  carrying  capacity  per  cable  vs.  number  of 
cables  per  duct  line  for  three  conductor  paper  and  lead  cables. 

cost  per  ampere  would  be  less  than  that  at  100  amp.  with  a 
20-year  life  for  the  cable.  If  a  cable  is  to  be  run  at  a  load  greater 
than  that  considered  as  perfectly  safe  from  a  standpoint  of  heat- 
ing, there  is  the  further  consideration  of  increased  dielectric 
losses  and  increased  resistance  of  conductor.  The  problem  has 
great  possibilities  when  more  data  becomes  available. 

Figure  51  also  shows  how  the  most  economical  load  increases 
as  the  load  factor,  and  hence  the  value  of  equivalent  hours, 
decreases.  Since  cable  heating  has  somewhat  of  a  cumulative 
effect,  it  is  generally  possible  to  carry  higher  loads  at  low  than  at 
high  load  factors.  The  safe  load  will,  therefore,  bear  somewhat 
the  same  relation  to  the  most  economical  load,  in  any  case. 
Hence  it  appears  that,  in  most  conditions  met  with  in  practice, 


UNDERGROUND  LINES  187 

the  load  carried  will  be  governed  by  the  capacity  of  the  cable 
rather  than  by  economy  as  usually  considered,  although,  of 
course,  economy  is  realized  by  not  overloading  a  cable  to  the 
point  of  injury. 

Economical  Route. — The  most  economical  route  for  an  under- 
ground line  may  be  an  important  consideration  in  its  layout. 
On  account  of  the  high  construction  cost,  the  use  of  the  shortest 
possible  route  is  even  more  important  in  this  case  than  with 
overhead  lines.  Other  things  being  equal,  this  might  often  point 
to  the  use  of  private  right-of-way.  The  relative  inaccessibility 
of  such  lines  on  private  property,  the  possibility  of  interference  by 
future  buildings,  etc.,  unless  the  property  is  bought  outright, 
and  the  difficulty  of  draining  manholes  in  many  cases,  generally 
makes  it  preferable  to  keep  such  lines  on  the  public  highway. 
The  number  and  location  of  manholes  necessary  may  have 
considerable  bearing  on  the  choice  of  a  route.  In  runs  of  only 
a  few  ducts,  the  cost  of  manholes  is  a  large  proportion  of  the  total 
and  hence  is  relatively  very  important.  In  runs  with  a  large 
number  of  ducts  the  manhole  cost  is  less  important  compared 
with  the  total  cost  of  the  line,  but  in  any  case,  the  route  requiring 
the  fewest  manholes,  other  things  being  equal,  has  considerable 
advantage. 

Secondary  Distribution. — The  problem  of  underground  sec- 
ondary distribution  is  quite  different  from  that  on  overhead 
lines.  Transformer  locations  are  limited  to  the  manholes  which 
in  turn  must  be  spaced  for  convenience  in  installing  and  main- 
taining the  cable.  The  secondaries  and  services  must  be  designed 
to  transmit  the  required  load  with  ample  provision  for  future 
contingencies.  The  changing  of  secondaries  of  services  or  of 
transformer  locations  is  a  considerably  more  serious  matter  than 
on  overhead  lines.  In  general  the  design  of  an  underground 
secondary  system  is  a  problem  requiring  a  great  deal  of  care  and 
good  judgment  in  predicting  future  loads  and  providing  for 
them  in  an  economical  manner. 

Number  of  Ducts  in  a  Duct  Line. — There  are  a  number  of 
problems  relating  to  the  construction  and  arrangement  of  ducts 
and  cables  which  are  worthy  of  attention.  An  example  will  be 
given  here  of  a  method  of  determining  the  most  economical 
number  of  ducts  which  should  be  placed  together  in  one  duct 
run.  While  it  would  not  always  be  practicable  to  limit  the 
number  of  ducts  to  the  figures  which  might  result  from  such  a 


188  ECONOMICS  OF  ELECTRICAL  DISTRIBUTION 

study,  the  knowledge  thus  gained  would  at  least  be  an  aid  in 
making  the  choice.  For  simplicity,  in  this  example  it  will  be 
assumed  that  the  ducts  are  all  filled,  the  cables  are  all  of  the  same 
size,  and  the  loads  carried  on  all  the  cables  are  of  equal  amounts 
and  identical  characteristics.  Naturally  such  a  condition  would 
rarely  be  found  in  practice.  However,  if  the  method  of  attacking 
the  problem  is  once  established,  it  can  be  extended  to  cover  other 
cases  of  dissimilar  cables  and  loads. 

The  basis  for  determining  economical  conditions  is  the  annual 
cost  per  ampere  transmitted  over  the  line  as  a  whole.  The 
equation  for  annual  cost  of  the  line  is  made  up  as  follows : 

Annual  cost  per  1,000  ft.  =  g  (cost  of  1,000  ft.  of  duct  line,  of  n 

conduits,  in  place) 
+  g    (average  cost  per   1,000     ft.   for 

manholes    complete    with    sewer 

connections) 
+  g   (cost   per   1,000   ft.   of   n   cables 

installed) 
-1-  cost  of  energy  losses  per  year  over 

1,000  ft.  of  n  cables.  (76) 

Cost  of  Duct  Line. — The  cost  of  installing  a  duct  line  is  made  up 
of: 

1.  Cost  of  excavating,  backfilling,  paving,  etc.  which  is  nearly 
proportional  to  the  width  of  the  duct  line.     If  the  cross-section  of 
the  duct  is  square,  or  nearly  so,  the  width  is  practically  propor- 
tional to  the  square  root  of  the  number  of  conduits,  \/n. 

2.  Cost  of  materials  used  and  labor  of  installing  which  is 
practically  proportional  to  the  number  of  conduits,  n. 

3.  Cost  of  transportation,  tools,  water  connections,  etc.  which 
is  practically  the  same  for  all  sizes. 

The  total  cost  of  the  duct  in  place  is  then  represented  approxi- 
mately by  the  expression  KI  +  K2  \/n  +  K3  n  where  K\,  Kz  and 
K3  are  constants  to  be  determined  from  actual  field  costs  on 
several  jobs. 

Cost  of  Manholes. — The  cost  of  building  a  standard  manhole 
is  usually  easy  to  determine.  The  cost  of  drainage  and  sewer 
connections  will  vary,  of  course,  with  different  conditions  en- 
countered. An  average  figure  must  be  assumed.  By  deter- 
mining the  average  number  of  manholes  per  1,000  ft.  of  duct,  the 


UNDERGROUND  LINES 


189 


annual  charges  per  1,000  ft.  for  manholes  complete  may  be 
easily  computed. 

Cost  of  Cables. — The  cost  per  foot  of  cable  is  readily  deter- 
mined from  current  prices  and  average  labor  costs  of  the  system 
under  consideration. 

Energy   Losses. — The    energy    loss    depends   on   the   current 


4000 


2         4         6        8        10        fc        14       16 
Ducts  and  Cables  per  Duct  Line 

FIG.  53. — Curves  of  total  current  carrying  capacity  of  duct  lines  vs.  number  of 
ducts  for  various  3  conductor  paper  and  lead  cables. 

carried.  If  the  number  of  conduits  in  the  duct  is  large,  the 
allowable  current  per  cable  will  be  smaller  than  in  a  duct  run  of 
few  conduits  on  account  of  the  increased  heating  effect.  A 
study  of  the  current  carrying  capacities  of  lead  covered  cables 
under  various  conditions  is  given  by  Ralph  W.  Atkinson,  in 
the  Journal  of  the  A.  I.  E.  E.  for  September,  1920.  By  use  of 
the  figures  and  charts  given  in  that  paper,  the  variation  of  the 
allowable  current  on  a  cable  with  the  number  of  conduits  in  the 
duct  line  was  derived,  for  three  types  of  cables,  and  plotted  in 
Fig.  52.  Figure  53  is  derived  from  Fig.  52  and  gives  the  total 
allowable  number  of  amperes  carried  on  the  duct  line  as  a  whole, 
for  any  number  of  conduits. 

If  Ic  =  the  allowable  current  per  cable, 

and  r  =  the  resistance  per  1,000  ft.  of  one  conductor  of  one 
cable.  The  annual  cost  of  energy  losses  per  1,000  ft. 

=  »  (3J.V  X  i  X  365  X  j^) 


190 


ECONOMICS  OF  ELECTRICAL  DISTRIBUTION 


g. — The  value  of  g,  i.e.,  the  per  cent  of  interest,  taxes,  depre- 
ciation, etc.  will  vary  for  the  different  classes  of  material.  Such 
figures  as  the  following  can  be  assumed  for  any  problem. 

Interest 7  per  cent 

Taxes 2  per  cent 

Depreciation,  etc.,  on  ducts 2  per  cent 

on  manholes 5  per  cent 

on  cables 5  per  cent 

Total  Annual  Cost. — After  the  various  individual  costs,  as 
indicated  above,  have  been  evaluated,  they  may  be  combined  to 
give  the  total  annual  cost.  Figures  54,  55,  and  56  show,  for 


6         8         10        12       14-       16 
Ducts  and  Cables 


FIG.  54. — Curves  showing  annual  costs  per  1,000  feet  vs.  number  of  ducts  and 
cables  for  No.  00  3-conductor  cables  (23,000  volts). 

the  three  different  types  of  cable,  the  variation  of  the  cost  of 
the  different  items  given  above  as  a  function  of  the  number  of 
ducts  in  the  line.  The  numerical  values,  of  course,  apply  only  to 
the  local  conditions  for  which  they  were  derived.  The  curve  for 
total  annual  cost  is  a  summation  of  the  separate  charges.  If  the 
total  annual  cost  for  any  given  number  of  ducts  and  cables  is 
divided  by  the  total  allowable  current  carried,  as  given  by  Fig. 
53,  the  cost  per  ampere  is  obtained.  This  is  given  in  the  upper 
curve  on  Figs.  54,  55  and  56. 

Results  Shown. — The  curves  indicate  clearly  the  high  cost  of 
less  than  four  ducts  in  a  run.     For  larger  lines,  the  point  of  econ- 


UNDERGROUND  LINES 


191 


omy  is  not  so  plainly  shown.     There  appears  to  be  comparatively 
little  difference  in  economy  between  duct  runs  of  from  4  to  16 


10,000 


4-        6        8        10       \1 
Ducts  and  Cables 

FIG.  55. — Curves  showing  annual  costs  per  1,000  feet  vs.  number  of  ducts  and 
cables  for  450  cm.  3-conductor  cables  (4,600  volts). 


^        4        G         8         10        12.       14       16 
Ducts  and  Cables 

FIG.  56. — Curves  showing  annual  costs  per  1,000  feet  vs.  number  of  ducts  and 
cables  for  200  M  cm.  three  conductor  cables  (2,300  volts). 

ducts.  The  curves  for  2,300- volt  cable,  Fig.  56,  indicates  a 
minimum  point  at  about  eight  ducts,  but  the  curve  is  fairly  flat. 
The  reason  for  the  small  difference  in  cost  shown  is  due  to  the  fact 


o 


o 


o 


192  ECONOMICS  OF  ELECTRICAL  DISTRIBUTION 

that  as  the  number  of  ducts  increases  the  allowable  current 
per  cable  decreases.  It  would  seem  to  be  indicated,  in  this  case, 
that  there  would  be  considerable  advantage  in  not  building  duct 
runs  to  provide  too  far  into  the  future.  If  the  total  cost  per 
ampere  of  a  six-  or  eight-duct  run  is  no  more  than  that  of  a  16, 
it  would  be  better  to  build  the  smaller  size  and  when  that  is  filled, 
build  another  of  the  same  size,  thus  saving  the  investment  on 
empty  ducts  for  a  considerable  period.  Of  course,  if  different 
cost  figures  were  used  or  different  assumptions  as  to  cable  sizes 
and  loadings  were  made,  the  points  of  greatest  economy  might  be 
more  pronounced.  Hence  these  curves  must  be  considered  as 
examples  of  the  method  only  and  not  for  general  application. 

Arrangement  of  Ducts. — Another  detail  of  construction  which 
might  offer  a  profitable  field  for  investigation  is  the  arrangement 
of  ducts  in  a  duct  run.  It  is  realized  that  the  practical  difficulties 
in  construction  may  be 'the  deciding  factor  in 
this  matter.  However,  it  must  be  kept  in 
mind  that  the  carrying  capacity  of  a  cable 
depends  a  great  deal  upon  its  location  with 
respect  to  other  cables.  The  center  cable  in 
a  nine  duct  run  arranged  in  a  square  will  have 
considerably  less  capacity  than  the  outside 
FIG.  57.— Arrange-  cables,  while  the  corner  cables  will  have  more 
ment  of  ducts  in  duct  capacity  than  those  between  (Fig.  57) .  This 
is  due  to  the  relative  capability  of  heat  disper- 
sion of  the  various  positions.  It  would  be  an  interesting  problem 
to  determine  how  much  extra  expense  would  be  justifiable  in 
order  to  use  some  arrangement  which  would  accomplish  better 
heat  dispersion.  The  problems  of  increasing  the  current  capacity 
of  a  duct  line  by  using  the  center  ducts  for  some  form  of  a  cool- 
ing system  or  of  flooding  the  runs  are  similar.  No  definite  data 
is  at  present  available  on  these  questions. 

Arrangement  of  Cables  in  a  Duct  Line. — The  question  of  the 
relative  heating  of  cables  leads  to  the  problem  of  the  proper 
arrangement  of  cables  in  a  given  duct  run.  It  rarely  happens 
that  the  cables  are  all  of  the  same  size  or  kind  or  carry  similar 
loads.  The  diversity  between  loads  may  have  considerable 
bearing  on  the  most  economical  arrangements  of  the  cables. 
Obviously  a  cable  carrying  a  heavy  load  with  a  high-load  factor 
can  operate  more  efficiently  if  placed  in  an  outside  duct  where  the 
heat  can  be  more  rapidly  dissipated.  Similarly,  if  a  cable  carry- 


o 


o 


o 


o 


o 


o 


UNDERGROUND  LINES  193 

ing  night  lighting  load  only  is  placed  in  an  interior  duct,  sur- 
rounded by  ducts  containing  cables  with  day  power  loads  only, 
and  the  peak  loads  do  not  overlap,  the  lighting  cable  can  be 
operated  at  a  higher  load  than  would  be  considered  safe  with  all 
cables  similarly  loaded. 

Possible  Savings  Large. — The  possibilities  for  economic 
investigations  on  underground  lines  are  large.  The  loads  carried 
are  fairly  heavy  as  a  rule,  since  underground  work  is  usually  done 
in.  congested  districts.  The  construction  cost  is  high.  Savings 
of  a  few  per  cent  mean  a  relatively  large  amount  of  money  in  the 
long  run.  The  examples  presented  above  and  the  other  problems 
mentioned  are  intended  merely  as  suggestions  for  study  along 
this  line.  The  available  data  on  heating  of  cables,  allowable 
current  carrying  capacity,  ageing,  etc.  is  as  yet  so  unreliable  that 
conclusive  solutions  of  any  problem  are  difficult. 


CHAPTER  XVI 
THE  SYSTEM  AS  A  WHOLE 

There  still  remain  a  great  many  problems,  both  large  and  small, 
which  are  continually  confronting  engineers  in  charge  of  distribu- 
tion systems.  Some  of  these  are  special  cases  but  can  be  solved 
by  an  adaptation  of  the  principles  indicated  herein.  Others 
apply  to  other  parts  of  the  system  than  the  distribution  lines, 
such  as  to  details  in  the  design  of  the  generating  station  or  the 
substations.  No  attempt  has  been  made  to  cover  such  questions 
specifically,  although  the  general  methods  could  be  applied. 
There  are  still  other  problems  which  deal  with  the  system  as  a 
whole.  While  it  is  not  within  the  province  of  this  work  to  discuss 
such  questions  at  length,  a  few  of  the  most  important  will  be 
mentioned  here. 

A  considerable  amount  has  been  published  at  different  times 
and  at  various  places  about  the  proper  location  of  a  generating 
station.  Theoretically,  it  should  be  located  in  such  a  way,  with 
respect  to  the  loads  to  be  carried,  that  the  total  annual  cost  on 
the  completed  system  will  be  a  minimum.  This  does  not 
necessarily  mean  that  the  best  location  is  at  the  center  of  gravity 
of  the  loads  as  is  sometimes  stated.  If  we  consider  two  equal 
loads,  one  operating  24  hr.  per  day  and  the  other  1  hr.  per  day,  it 
is  obvious  that  the  generating  station  should  not  be  half  way 
between  but,  rather,  nearer  the  load  with  the  high  load  factor. 
From  a  practical  standpoint,  the  location  theoretically  best  is 
rarely  attainable.  There  are  many  other  important  considera- 
tions such  as  transportation  facilities  for  fuel,  available  supply  of 
cooling  water,  practicable  building  sites,  etc.,  which  govern  the 
choice.  It  might  be  said  that,  with  the  present-day  design  of 
stations  and  the  large  capacities  being  attained,  the  matter  of 
available  water  supply  is  becoming  all  important.  The  question 
of  generating  energy  at  the  mouth  of  coal  mines  and  transmitting 
to  distant  points  has  been  much  talked  about  recently.  Such  a 
practice  seems  at  present,  to  be  limited,  however,  to  exceptional 
cases  where  an  ample  water  supply  may  be  had.  In  such  cases 

194 


THE  SYSTEM  AS  A   WHOLE  195 

economy  is  realized  if  the  energy  can  be  so  generated  and  trans- 
mitted to  the  point  of  utilization  at  a  less  cost  per  year,  including 
investment  charges  and  cost  of  losses  on  transmission  lines,  than 
the  cost  of  the  same  energy  generated  at  the  feeding  point,  includ- 
ing the  cost  of  transporting  the  coal  by  freight.  In  any  case  the 
economy  of  any  location  for  a  generating  station  is  determined 
only  by  a  complete  study  of  the  annual  costs  of  all  alternatives. 

The  location  of  a  substation  is  a  somewhat  similar  problem. 
In  this  case,  however,  the  limiting  factors  are  usually  not  so 
many,  and  the  choice  may  be  made  on  a  more  theoretical  basis. 
A  careful  consideration  of  the  loads  to  be  carried,  the  lengths  of 
the  necessary  lines,  both  underground  and  overhead,  and  the 
energy  losses  involved,  will  usually  be  very  profitable.  It  often 
occurs  that  the  expenditure  of  a  little  more  money  in  the  purchase 
of  the  best  possible  site  may  be  repaid  several  times  over  in  the 
economies  effected  on  the  distribution  system. 

The  previous  chapters  in  this  book  have  discussed  the  problems 
pertaining  to  each  part  of  the  system  with  very  little  regard  for 
the  relation  of  that  part  to  the  system  as  a  whole.  If  the  study 
of  the  economics  of  the  distribution  system  is  to  be  made  com- 
plete, this  inter-relation  must  be  considered.  For  example, 
we  may  determine  the  most  economical  drop  in  voltage  on  a 
transmission  line  and  also  on  a  power  line.  However,  in  order  to 
properly  serve  the  customer,  the  voltage  regulation  must  be 
maintained  to  a  certain  standard.  The  problem  is,  then,  to 
determine  the  most  economical  design  and  arrangement  of  trans- 
mission lines,  substation,  power  lines  and  regulators  to  accomplish 
the  desired  result.  This  may  possibly  be  somewhat  different 
than  the  solutions  for  the  individual  parts.  Such  a  study 
involves  a  careful  determination  of  annual  charges  on  all  types  of 
construction  and  a  complete  computation  of  the  cost  of  energy 
loss. 

Immediate  results  should  not  necessarily  be  expected  from  an 
application  of  engineering  economics  to  an  established  dis- 
tribution system.  In  some  cases  the  money  saved  on  one  or  two 
lines  may  prove  the  work  well  worth  while.  In  other  cases,  the 
results  will  be  evident  only  after  some  length  of  time.  It  is 
rare  that  a  system  can  be  brought  up  to  an  economical  standard 
in  a  short  time.  Usually  the  engineer  must  be  content  to 
improve  conditions  gradually,  by  designing  new  extensions  and 
rebuilding  old  work,  here  and  there,  in  accordance  with  economi- 


196  ECONOMICS  OF  ELECTRICAL  DISTRIBUTION 

cal  principles.  The  final  results  will  be  shown  in  the  improved 
efficiency  on  the  system  as  a  whole.  Often,  this  will  take  the 
form  of  a  reduction  in  overall  losses  on  the  system,  although  this 
is  not  necessarily  the  case.  The  economical  percentage  of  loss 
will  depend  on  the  relation  between  construction  costs  and  the 
cost  of  energy,  which  may  be  very  different  for  different  systems. 
Hence  the  percentage  of  energy  lost  is  not  a  true  measure  of 
economy.  The  ideal  condition  of  maximum  efficiency  to  which 
it  is  the  purpose  of  all  economic  study  to  contribute,  is  that 
conditions  in  which  every  customer  is  provided  with  a  reasonably 
good  quality  of  service  at  the  least  possible  cost  over  the  whole 
system. 


CHAPTER  XVII 
INDUSTRIAL  PLANT  PROBLEMS 

APPLICATION  TO  INDUSTRIAL  PLANT  PROBLEMS  OF  THE 

PRINCIPLES  OF  ECONOMICS  OUTLINED  FOR  ELECTRICAL 

DISTRIBUTION 

This  book  has  dealt  primarily  with  electrical  distribution  prob- 
lems from  the  point  of  view  of  the  central  station.  The  purpose, 
in  general,  has  been  to  indicate  means  of  studying  a  distribution 
system  with  the  view  to  transmitting  electrical  energy  from  the 
generating  plant  to  the  customer  at  the  least  possible  over-all 
cost.  The  second  part  of  the  book  so  far  has  dealt  entirely  with 
the  problems  encountered  in  the  various  parts  of  such  a  distribu- 
tion system.  The  consumer,  however,  is  interested  but  indirectly 
in  such  problems,  in  that  a  reduction  in  central  station  costs  may 
lead  to  lower  rates  or  better  service.  The  consumer  of  any 
considerable  amount  of  energy,  however,  such  as  a  large  indus- 
trial plant,  has  numerous  problems  of  his  own  which  may  be 
classed  as  problems  of  electrical  distribution.  These  may  be 
viewed  from  the  standpoint  of  the  producer  or  the  buyer  of 
electrical  energy  according  to  whether  he  produces  his  own  energy 
or  buys  it  from  the  central  station.  In  some  cases  these  problems 
are  very  similar  to  those  of  the  central  station.  In  other  cases 
they  may  be  quite  different.  In  any  case,  however,  where  the 
question  of  economy  enters,  the  principles  explained  in  Part  I 
will  be  found  fundamental.  Annual  cost  is  the  basis  of  compari- 
son for  any  alternative  propositions.  In  general,  the  solution 
of  many  of  the  problems  may  be  simplified  by  the  use  of  a  general 
equation  such  as  that  described  -in  Chapter  VI.  The  most 
economical  condition  will  be  discovered,  only  if  all  items  of 
expense  including  the  cost  of  energy  losses  as  well  as  the  fixed 
charges  on  investment  are  considered. 

Voltage  Regulation. — The  problem  confronting  the  electrical 
engineer  in  an  industrial  plant  is,  essentially,  the  same  as  that  of 
the  distribution  engineer  as  stated  in  Chaper  I,  i.e.,  to  realize 
the  greatest  economy  possible  consistent  with  good  service.  In 
this  case  good  service  depends  upon  two  factors.  In  the  first 

197 


198  ECONOMICS  OF  ELECTRICAL  DISTRIBUTION 

place,  the  central  station  must  furnish  reasonably  good  regulation 
at  the  consumer's  service.  This  is  usually  more  or  less  regulated 
by  contract  but  may  depend  somewhat  on  the  character  of  load 
imposed  by  the  consumer  as  to  power  factor,  fluctuation,  etc. 
A  customers  load  may  be  of  such  a  nature  that  it  is  practically 
impossible  to  give  good  regulation  at  his  service  and  other  services 
on  the  same  line  may  be  similarly  disturbed.  In  such  a  case  it  is 
usually  necessary  for  the  customer  to  change  his  equipment  or 
method  of  operation  in  some  way  so  as  to  remedy  the  difficulty. 
The  second  factor  is  the  interior  distribution  of  the  plant  itself. 
Assuming  that  good  regulation  is  furnished  by  the  central  station, 
it  is  essential  that  the  arrangement  of  the  plant  electrical  dis- 
tribution be  such  that  good  voltage  conditions  are  maintained  at 
the  points  of  utilization.  By  plant  distribution  is  meant  all 
parts  of  the  electrical  circuit  including  that  of  apparatus  of 
utilization.  All  such  problems  may  at  first  be  more  electrical 
than  economic.  However,  good  service  being  accomplished,  it  is 
then  essential  to  investigate  the  matter  of  economy. 

Problems  of  Power  Distribution. — The  problems  pertaining 
to  the  actual  wiring  in  an  industrial  plant  will  be  very  similar  to 
those  of  the  larger  distribution  system  which  has  been  heretofore 
described.  For  small  plants  it  will  be  simply  a  case  of  secondary 
distribution.  The  sizes  of  conductor  can  be  readily  determined 
by  a  consideration  of  the  annual  charges  on  the  cost  of  the  con- 
ductor in  place  versus  the  cost  of  losses.  In  the  case  of  the 
consumer,  the  unit  cost  of  lost  energy  is  more  easily  determined 
than  for  the  central  station  as  it  appears  very  decidedly  in  his 
monthly  bills.  For  larger  plants,  with  several  separate  buildings, 
it  may  be  advisable  to  distribute  partly  at  primary  voltage. 
For  still  larger  ones,  generating  their  own  power,  the  problem 
of  high  voltage  transmission  may  enter.  In  any  such  case  the 
problems  of  the  central  station  are  more  nearly  approached  and 
the  examples  cited  in  the  preceding  chapters  will  apply. 

In  many  plants  the  question  arises  as  to  the  comparative 
advantages  of  electrical  distribution  of  power  as  compared  with 
mechanical,  i.e.,  individual  motors  on  every  machine  instead  of 
large  motors  with  mechanical  transmission  to  a  group  of  machines. 
Of  course,  there  are  many  factors  in  such  a  problem,  including 
the  type  of  machines  used  and  method  of  operation.  In  general 
a  study  of  the  total  annual  cost  of  operation  including  fixed 
charges,  electrical  and  mechanical  losses,  and  a  consideration 


INDUSTRIAL  PLANT  PROBLEMS  199 

of  any  possible  difference  in  production,  labor  costs,  maintenance, 
etc.  will  be  of  great  value  in  determining  the  proper  choice.  Such 
a  study  will  be  based  on  the  principles  explained  in  Part  I. 

Problems  of  Equipment. — The  choice  of  proper  equipment  may 
have  considerable  effect  on  the  economy  of  plant  operation.  For 
customers  buying  energy  at  primary  voltage  there  is  first  the 
selection  of  proper  primary  switch  house  equipment.  Trans- 
formers must  be  selected  with  the  view  of  probable  increase  in 
load.  However,  if  too  large  units  are  installed,  not  only  is  there 
a  waste  of  investment  but  the  additional  core  losses  and  the  effect 
on  the  power  factor  may  be  important.  It  will  sometimes  be 
found  economical  in  such  a  case  to  use  an  open  delta  installation 
until  the  load  justifies  the  installation  of  the  third  unit.  Where 
reserve  transformers  are  required  in  case  of  trouble  where  a 
shutdown  would  be  serious,  the  choice  of  the  size  of  units  will 
be  influenced  by  that  fact.  In  any  such  case  as  well  as  in  the 
selection  of  other  equipment  such  as  switches,  etc.,  a  study  of  the 
total  annual  cost  is  essential. 

A  very  common  fault  in  the  selection  of  apparatus  of  utilization 
is  that  of  overmotoring.  Where  careful  engineering  has  not  been 
done,  the  tendency  often  is  to  install  larger  motors  than  are  neces- 
sary for  the  use  required  with  the  idea  that  the  reserve  power  ob- 
tained is  advantageous.  The  investment  charges  in  such  a  case 
are  larger  than  necessary.  Also,  any  considerable  underloading 
of  induction  motors  brings  down  the  power  factor  and  this  may 
have  considerable  effect  on  the  regulation.  In  case  poor  power 
factor  is  penalized  in  the  rates  it  is  important  to  maintain  as 
good  a  value  as  possible.  A  careful  study  of  the  diversity  factor 
between  machines  on  the  same  motor  is  important  in  its 
proper  size.  In  some  cases  synchronous  motors  operated  to  im- 
prove power  factor  or  static  condensers  will  be  found  advisable. 
Similar  considerations  will  be  encountered  with  other  types  of 
equipment.  Old  inefficient  equipment  may  often  be  replaced 
to  advantage,  although  it  must  be  kept  in  mind  that  beyond  a 
certain  point  efficiency  is  not  always  economy.  In  all  cases,  the 
annual  cost  must  always  be  considered  as  an  important  factor 
along  with  probably  load  increase,  regulation,  reserve  power,  etc. 

Problems  of  Operation. — In  this  country  the  establishment  of 
rates  modified  by  power  factor  considerations  is  becoming  more 
and  more  important  and  is  at  present  receiving  considerable 
attention  from  the  central  stations.  Poor  power  factor  not  only 


200  ECONOMICS  OF  ELECTRICAL  DISTRIBUTION 

occasions  additional  power  losses  all  the  way  back  to  the  genera- 
tors but  it  increases  the  difficulties  of  maintaining  good  regula- 
tion. It  would  therefore  seem  just  that  a  customer  with  a  low 
power  factor  should  pay  more  per  kilowatt  hour  than  one  with  a 
high  power  factor.  With  this  in  mind  it  is  essential  that  every 
user  obtain  as  good  a  power  factor  as  possible  from  his  plant. 
This  may  be  done  by  a  selection  of  equipment  of  proper  size  and 
type  for  the  load  as  mentioned  above  and  also  by  the  use  of 
condensers,  either  static  or  synchronous.  The  expenditure 
justified  in  order  to  increase  power  factor  can  be  determined  by 
a  study  of  the  reduction  in  the  power  bills  so  effected.  In  this 
way  it  will  be  discovered  whether  it  will  be  a  paying  proposition 
to  establish  a  power  factor  of  say  95  per  cent  or  of  80  per  cent,  etc. 

The  plant  load  factor  will  usually  be  an  important  considera- 
tion in  a  study  to  increase  economy.  At  present  most  power 
rates  are  based  on  some  form  of  demand  charge  and  some  form 
of  kilowatt  hour  charge.  It  is  obviously  advantageous  to  reduce 
the  demand  charge  if  possible,  spreading  the  energy  used  at  the 
peak  through  the  rest  of  the  day.  This  is  especially  true  where 
the  rates  are  of  such  a  form  as,  the  demand  load  for  "a"  hours  at 
"b"  cents  per  unit  and  the  reminder  at  "c"  cents.  The  demand 
may  often  be  reduced  by  a  careful  study  of  the  time  of  starting 
up  motors  and  of  overlapping  operation  of  various  equipment. 
In  some  cases  the  reduction  in  the  annual  bills  will  warrant  the 
purchase  of  new  equipment  to  reduce  the  demand. 

Other  Problems. — It  is  not  within  the  province  of  this  book  to 
attempt  to  discuss  in  any  detail  all  problems  in  electrical  distribu- 
tion which  occur  in  an  industrial  plant.  It  has  been  attempted 
above  to  outline  some  of  the  questions  encountered  and  give  a 
general  idea  of  the  field  of  application  of  economic  principles  to 
such  problems.  There  are  many  others  which  are  less  strictly 
electrical  or  not  at  all  electrical  which  will  bear  a  similar  study. 
The  questions  of  proper  illumination  as  affecting  production, 
safety  devices,  labor  saving  devices,  etc.  are  all  more  or  less 
economic  problems.  The  main  point  must  be  always  kept  in 
mind  that,  once  good  service  is  accomplished,  the  greatest  advan- 
tage lies  with  any  installation  for  which  the  total  annual  cost, 
everything  considered,  is  less  than  any  other. 


APPENDIX  A 
METHOD  OF  APPROXIMATING  ENERGY  COST 

It  very  often  happens  that  it  is  desirable  to  study  the  economics 
of  different  types  of  circuits  before  it  has  been  possible  to  under- 
take any  detailed  determination  of  energy  cost  as  outlined  in 
Chap.  IV.  It  has  been  customary,  with  most  engineers  doing 
such  work,  to  assume  in  such  a  case,  an  approximate  average 
figure  such  as  .01  per  kilowatt-hour  or  some  other  figure  which  is 
known  to  be  somewhere  near  the  average  for  the  whole  output 
of  the  system.  A  little  closer  approximation  may  be  made, 
keeping  the  principles  set  forth  in  Chap.  IV  in  mind,  by  determin- 
ing roughly  the  variation  in  cost  with  the  load  factor  for  the  class 
of  load  being  considered.  While  not  very  accurate,  such  figures 
are  at  least  better  than  a  rough  guess  of  a  cost  to  cover  all  cases. 

An  example  of  such  a  determination  will  be  worked  out  here  to 
show  how  it  is  possible  to  handle  such  a  problem.  The  general 
methods  used  may  be  adopted  to  other  similar  cases. 

The  loads  considered  are  power  loads  in  the  suburban  dis- 
tricts, i.e.,  such  that  considerable  transmission-line  cost  is 
involved.  As  was  explained  in  Chap.  IV,  the  demand  cost  for 
energy  losses  is  very  nearly  proportional  to  the  amount  of  load 
at  time  of  station  peak.  It  was  assumed  that  the  average 
demand  cost  at  the  generating-station  switchboard  equals  $15.76 
per  kilowatt  and  the  average  kilowatt-hour  cost  =  $  .0036  per 
kilowatt-hour.  (These  costs  are  usually  available  and  fairly 
correct  in  most  companies.)  The  demand  cost  per  kilowatt-hour 
will  be  inversely  proportional  to  the  load  factor,  considering  load 
at  generating  station  only.  Reducing  the  demand  cost  to  a  cost 
per  kilowatt-hour: 

At  100  per  cent  load  factor,  demand  charge  = 


.0018  per  kilowatt-hour. 
For  any  other  load  factor,  demand  charge     =  .0018 

Load  factor  (ex- 

pressed as  a  fraction  not  as  per  cent.) 

In  order  to  determine  the  energy  cost  at  any  point  on  the 

201 


202  ECONOMICS  OF  ELECTRICAL  DISTRIBUTION 

system  beyond  the  generating-station,  the  costs  given  above 
must  be  increased  by  the  cost  of  transmission  and  distribution. 
Also  diversity  factors  must  be  taken  into  account. 

A  study  of  the  property  classification  of  the  system  indicated 
a  division  approximately  as  follows: 

PER  CENT 

Generating  plant 40 . 0 

Underground  conduit,  etc 6.5 

Poles  and  fixtures 7.0 

Transmission  lines 24 . 5 

Distribution 19.3 

Transformers 2.7 

We  may  then  assume,  roughly,  that  the  demand  charge  exclusive 

of  the  generating  plant  is  60  per  cent  and  of  the  generating  plant 

40  per  cent  of  the  total.     For  a  load  of  100  per  cent  load  factor 

the  total  demand  charge  would  then  be  simply  = 

generating-station  demand  charge 

.40 

=  2.5  generating-station  demand  for  100  per  cent  load  factor. 

For  any  other  load  factor,  the  diversity  between  loads  will 
affect  the  cost.  A  kilowatt  load  at  the  customer  will  not  mean 
a  kilowatt  at  the  generator  unless  the  peaks  happen  to  coincide. 
The  diversity  factors  were  assumed  in  this  case  to  be  those  given 
in  the  " Standard  Handbook"  for  general  power  loads  as  follows: 

PER  CENT 

Between  transformers 74 

Between  power  lines 87 

Between  substations 91 

It  is  assumed  that  these  factors  hold  good  for  a  load  factor 
of  25  per  cent.  Then,  1  kw.  at  the  customer  becomes 

1  X  1       =1  kw.  at  the  transformer. 
.  74  X  1       =  .  74  kw.  on  the  power  line. 
.87  X  .74  =  .61  kw.  at  the  substation. 
.91  X  .61  =  .  586  kw.  at  the  generating-station. 

These  ratios  combined  with  the  percentages  of  the  total  prop- 
erty values  as  given  above,  give  the  approximate  share  of  each 
kilowatt  at  the  customer  in  the  total  demand  charge  for  the 
system.  For  example,  if  the  transformer  investment  is  2.7  per 
cent  of  the  total,  and  1  kw.  at  the  customer  represents  1  kw.  at 


APPENDIX 


203 


the  transformer,  that  kilowatt  should  take  2.7  per  cent  of  the 
total  demand  charge  for  its  share  in  the  transformer  investment. 
If  it  becomes  only  .586  kw.  at  the  generating  station  and  the 
generating  station  represents  40  per  cent  of  the  total  investment, 
the  kilowatt  in  question  should  be  charged  with  23.5  per  cent  of 
the  total  demand  charge  for  its  share  in  the  station.  In  order  to 
make  property  and  diversity  factor  classifications  coincide  and 
as  an  approximation,  it  was  assumed  that  the  investment  in 
distribution  and  poles  and  fixtures  could  be  taken  to  represent 
power  lines,  and  that  the  investment  in  transmission  and 
underground  could  be  applied  to  substations. 

Then  for  1  kw.  at  the  customer 


Per  cent  total  inve 

stment 

Share  in  total 
demand  cost 

On  transformer 

1  0    kw.  X  2  7 

2    7 

On  power  line  

.74kw.  X  (19.3  + 

7.0) 

19.5 

At  substation  
At  generating  station  

.64    kw.  X  (  6  .  5  + 
.  586  kw.  X  40 

24.5) 

19.8 
23.5 

Total  

65  5 

Then  if  total  demand  for  any  load  factor 

_  gen.-sta.  demand  for  that  load  factor 

.40 
The  demand  charge  for  1  kw.  at  the  customer  with  the  load  factor  assumed 


(25  per  cent) 


gen. -station  demand  at  25  per  cent  load  factor  X  0.655 

.40 
=  1.63  X  gen. -station  demand  at  25  per  cent  load  factor 


From  the  two  values  thus  obtained  for  100  per  cent  and  25  per 
cent  load  factors  and  the  point  for  load  factor  equals  zero,  the 
curve  shown  on  Fig.  58  was  plotted  showing  by  what  amount  the 
generating  station  demand  cost  at  any  load  factor  should  be 
multiplied  to  give  the  demand  charge  at  the  customers  in  question. 
Similarly  the  kilowatt-hour  charge  must  be  increased  to 
account  for  losses  in  transmission  and  distribution.  It  was 
assumed  that  the  total  loss  between  generating  station  and 
customer  would  be  not  far  from  30  per  cent  and,  for  the  loads 
considered,  it  was  assumed  that  it  would  be  accurate  enough  to 
increase  the  generating-station,  kilowatt-hour  charge  by  30  per 


204  ECONOMICS  OF  ELECTRICAL  DISTRIBUTION 

cent  to  give  the  charge  at  the  customer.     The  average  kilowatt- 
hour  charge  would  then  =  .036  X  1.36  =  .0047. 

The  following  table  may  then  be  derived.     The  first  column 
gives  the  average  number  of  hours  of  operation  per  week  (tw)  at 


~ent  Generating  Station  Demand  Charge  /io 
o  t-  -  r^  *• 

jt  O  W  O  u 

^ 

-^ 

X 

* 

/ 

' 

/ 

/ 

/ 

'0     O.I     02    0.5    OA    0.5     0.6    0.7    0.8     0-9     1.0      I.I     12 
Load  Factor 

FIG.  58. — Three-phase  secondary.  Factor  by  which  demand  charge  at  gene- 
rating station  for  any  load  factor  must  be  multiplied  to  obtain  demand  charge 
at  load. 


full  load  corresponding  to  any  load  factor.  The  last  column 
gives  the  corresponding  values  of  twCe.  These  are  used  as  shown 
in  Chaps.  XI  and  XIV.  The  values  of  twCe  are  plotted  in  Fig.  59 


0     10     ZO     30     40     50    60     70     80    90     100    110    120    130    140    ISO    160    170   180 
Hours  per  Week 

FIG.  59. — Values   of   twCe   for   different   values   of   tw  (hours  per  week) . 


The  cost  of  energy  obtained  in  this  way  is  very  approximate 
but  may  serve  the  purpose  until  some  more  accurate  determina- 
tion can  be  made.  The  figures  here  given  must  be  considered 
as  examples  only  and  not  as  representative  of  present-day  costs 
on  any  system. 


APPENDIX 
TABLE  20 


205 


*. 

Load 
factor 

Station 
demand 

Multi- 
plier 

Demand 
at  load 

Total  at 
load 

M7. 

.0 

.00 

.0 

1.00 

.00 

.0 

.0 

16.8 

.10 

.018 

1.28 

.0231 

.0278 

.467 

33.6 

.20 

.009 

1.52 

.0137 

.0184 

.619 

50.4 

.30 

.006 

1.72 

.0103 

.0150 

.756 

67.2 

.40 

.0045 

1.90 

.0086 

.0133 

.894 

84.0 

.50 

.0036 

2.05 

.0074 

.0121 

1.015 

100.8 

.60 

.003 

2.20 

.0066 

.0113 

1.139 

117.6 

.70 

.0026 

2.32 

.0060 

.0107 

1.258 

134.4 

.80 

.00225 

2.41 

.0054 

.0101 

1.358 

151.2 

.90 

.002 

2.47 

.0049 

.0096 

1.450 

168.0 

1.00 

.018 

2.50 

.0045 

.0092 

1.543 

INDEX 


Atkinson,  Ralph  W.,  189 
B 

"B",  59 

Backbone  transmission  lines,  71 

Balance  factor,  38 


Cost,  records,  11,  14 

unit,  12 

variation,  effect  of,  11 
Costs,  11 

classification  of,  23 

method  of  making  classification, 
25 


I) 


Cables,  arrangement  of  in  duct  line, 

192 

cost  of,  189 
Conductor    economy,    transmission 

line,  83 
Conductors,  cost  of,  75 

economical      size     for      power 

secondaries,  159 
size,  lighting  circuits,  132 
power  circuits,  111,  121 
transmission  line,  83 
underground,  184 
Copper  loss,  secondary,  156 

transformer,  156 
Core  loss,  transformer,  155 
Cost,  annual,  12,  18 

annual  on  physical  property,  18 
for    secondary    distribution, 

142 

total  for  secondary  line,  158 
consumers,  24 
demand,  24 

apportioning  of,  27 
at  any  point,  29 
data  necessary  for,  31 
of  energy,  22 

variations  in,  32 
first,  12 

kilowatt  hour,  27 
labor,  formulas  for,  14 
of  lost  energy,  32,  75,  118 
output,  24 

apportioning  of,  27 


Data,  empirical,  7 

exact,  7 

Demand  charge,  204 
costs,  24 

apportioning  of,  27 
at  any  point,  29 
data  necessary  for,  31 
factor,  38 

Density,  load,  140,  141 
Depreciation,  19,  20 
Distribution    of   load    over   several 

lines,  124 
of   power   in  industrial  plants, 

198 

secondary  lines  single  phase,  139 
lines    underground, 

187 

Diversity  factor,  38 
Doherty,  H.  L.,  23 
Duct  line,  cost  of,  188 
Ducts,  arrangement  of,  192 

economical    number    in     duct 
run,  187 


E 


Economics,  application  to  distribu- 
tion problems,  5 
Eisenmenger,  H.  E.,  26,  29 
Energy  charge,  204 
cost,  22 

method  of  approximating,  201 
variations  in,  32 
loss,  cost  of  underground,  189 
Equation,  general,  48,  50 


207 


208 


INDEX 


Equipment,  industrial  plant,  199 

Equivalent  hours,  39 

corrections  for,  44 
for  lighting  circuits,  40,  45 
for  power  circuits,  120 
for  various  loads,  45 
relation  between  load  factor 
and,  45 


Financial  conditions,  9 

G 

"g",  49 

Generating  station,  location  of,  194 

H 

Hopkinson,  Dr.  John,  23 

I 

Increase  in  capacity,  135 
Industrial  plant  problems,  197 
Insulators,  cost  of,  75 
Insurance,  19 
Interest,  18 


Kelvin's  law,  48 

Kilowatthour,  total  charge  per,  32 
cost,  27 


Lighting  circuits,  128 

Lightning  arresters,  cost  of,  75,  79 

Line  layouts,  points  to  consider  in,  9 

secondary,  annual  cost,  143 
Lines,  underground,  183 
Load,  characteristics,  37 

curves,  for  secondaries,  160 

density,  140,  141 


Load,  most  economical  for  transmis- 
sion lines,  84 
prediction  on  lighting  circuits, 

129 

Loading,  13 
Lost  energy,  cost  of,  32 

M 

Maintenance,  19 
Manholes,  cost  of,  188 

O 

Obsolescence,  20 

Operation    problems,    industrial 

plant,  199 
Output  cost,  24 

apportioning  of,  27 
Overhead  expense,  13 


Parts  of  system,  relation  to  whole,  9 
Patrolling,  cost  of,  76 
Power  circuits,  109 

two  in  place  of  one,  121 
voltage,  110 
distribution,    industrial    plant, 

198 
factor,  37 

improvement,  111 
loss  and  voltage  drop,  55 
secondaries,  174 

Predicting  load  on  residence  lighting 
circuits,  129 


R 


Rate  making,  cost  of  energy  for,  22 
Reconstructed  line,  annual  cost,  98 
total  investment  represented, 

96 
utilization   of   old   materials, 

99,  104 
division  between  parallel  lines,      Reconstruction  problems,  96 

126  Regulation,    improvement    by    two 

factor,  39  circuits,  124 

increase  in,  8  industrial   plant  voltage,    197 


INDEX 


209 


Regulators    with    lighting    circuits, 

use  of,  134 

Repair,  maintenance  and,  19 
Replacement  of  serviceable  line,  21 
Results,  presentation  of,  51 
Right-of-way,  cost  of,  74 
Route,  economical  underground  line, 

187 
most    economical    transmission 

line,  89,  107 


Secondaries,     economical     size     of 

power,  176 
line  cost  curves,  161 
load  curves  for,  160 
power,  174 

economy  of  instead  of  sepa- 

rate transformers,  175 
S0  for  scattered  load,  171 
Secondary  copper  loss,  156 

distribution,  annual  cost,  142 
single  phase,  139 
underground,  187 
line,  annual  cost,  143,  154 
transmission  lines,  81 
Service,  good,  7,  138 
Shortening  a  line,  economy  of,  100. 
Silver,  A.  E.,  73 

Span,  standard  transmission  line,  82 
Special  apparatus,  cost  of,  75 
Substation  structures,  cost  of,  75 
Substations,  location  of,  195 
Switches,  cost  of,  75,  79 
Symbols,  50 
System  as  a  whole,  the,  194 


Taxes,  18 

"t  C.",  117 

Towers,  cost  of,  74 

Transformers,   annual  cost  of,   142 


Transformers,  copper  loss,  156 
core  loss,  155 
cost  of,  75,  77,  79 
curves,  162 
replacing,  163 
fixed  charges  on,  157 
size,  147,  168 
spacing,  most  economical,  146, 

167 

Transmission  line,  backbone,  71 
problems,  71 
secondary,  81 


Underground  lines,  183 

Voltage,  drop,  58 

approximate      method      for 

secondaries,  64 
most  economical  for  second- 

aries>  144>  166 

P°wer  circuits,  111 

and  P°wer  loss,  55 
Power  circuits,  110 
regulation,  in  industrial  plants, 

197 
standard  for  transmission  lines 

^2 
underground  lines,  184 


W 


Waste  energy,  cost  of,  32 

Wire,  secondary,  fixed  charges  on, 

157 

size,  power  circuits,  111,  121 
S<£  lighting  circuits   econom- 

ical,  132,  149,  168 
transmission  lines,  83 
underground  lines,  184 


LD  2l-ioom.8>>34 


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